Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #144225 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #10320 |
| commutator, Frattini, cosocle | C2×#131 | C23 |
| prime radical | #585 | D4 |
| LCS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
548 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 548 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,9,15)(3,16,22,64)(4,23,28,57)(5,29,34,61)(6,35,39,13)(7,14,8,42)(10,43,47,18)(11,37,52,26)(12,32,55,33)(17,65,68,122)(19,71,36,63)(20,53,21,76)(24,80,74,123)(25,83,87,126)(27,90,40,79)(30,60,70,46)(31,94,98,92)(38,59,41,58)(44,104,107,124)(45,69,56,102)(48,111,106,127)(49,113,115,128)(50,88,51,89)(54,97,101,100)(62,72,121,67)(66,105,84,108)(73,117,75,118)(77,125,120,81)(78,99,119,86)(82,112,103,116)(85,91,96,114)(93,110,95,109);(1,3,9,22)(2,10,15,47)(4,24,28,74)(5,30,34,70)(6,36,39,19)(7,21,8,20)(11,48,52,106)(12,53,55,76)(13,56,35,45)(14,60,42,46)(16,61,64,29)(17,40,68,27)(18,32,43,33)(23,77,57,120)(25,84,87,66)(26,82,37,103)(31,95,98,93)(38,73,41,75)(44,51,107,50)(49,62,115,121)(54,118,101,117)(58,109,59,110)(63,92,71,94)(65,119,122,78)(67,88,72,89)(69,97,102,100)(79,108,90,105)(80,126,123,83)(81,99,125,86)(85,104,96,124)(91,112,114,116)(111,128,127,113);(1,4)(2,11)(3,17)(5,25)(6,37)(7,40)(8,27)(9,28)(10,44)(12,49)(13,57)(14,51)(15,52)(16,62)(18,66)(19,72)(20,74)(21,24)(22,68)(23,35)(26,39)(29,91)(30,81)(31,96)(32,99)(33,86)(34,87)(36,67)(38,88)(41,89)(42,50)(43,84)(45,108)(46,106)(47,107)(48,60)(53,116)(54,119)(55,115)(56,105)(58,79)(59,90)(61,114)(63,122)(64,121)(65,71)(69,104)(70,125)(73,103)(75,82)(76,112)(77,110)(78,101)(80,117)(83,94)(85,98)(92,126)(93,111)(95,127)(97,113)(100,128)(102,124)(109,120)(118,123)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fi,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^e=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 51423);