Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7396 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #353 |
| commutator | C2×M16 | C2×C8 |
| Frattini, cosocle | #2151 | C22 |
| socle | C2 | #7066 |
| prime radical | C23 | #350 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×C8 ◁ C22×C8 ◁ #2151 ◁ G | |
| LECS | G ▷ #2151 ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
95 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,5,11,6,13,29,45,7,14,30,47,31,50,82,102,8,12,28,46,32,48,81,101,22,49,62,85,72,103,113,121)(3,15,17,53,19,55,64,106,20,56,66,107,69,109,116,128,18,54,65,98,67,108,115,125,68,38,27,78,79,91,80,120)(4,10,23,39,25,40,73,92,26,42,74,93,77,95,118,124,24,41,75,44,76,94,119,99,57,51,104,100,110,105,127,126)(9,33,34,83,35,84,87,60,36,63,88,112,89,114,123,21,37,70,86,71,90,117,122,16,52,58,43,59,96,111,97,61);(1,3)(2,9)(4,21,24,61)(5,27,28,66)(6,19)(7,20)(8,18)(10,38,41,56)(11,43,46,88)(12,37)(13,35)(14,36)(15,51,54,42)(16,57,60,26)(17,62,65,30)(22,68)(23,58)(25,71,76,83)(29,80,81,116)(31,69)(32,67)(33,74)(34,85,86,47)(39,53)(40,91,94,109)(44,98)(45,97,101,123)(48,90)(49,52)(50,89)(55,105,108,95)(59,110,112,77)(63,75)(64,113,115,82)(70,104)(72,79)(73,111)(78,100)(84,118)(87,121,122,102)(92,106)(93,107)(96,103)(99,125)(114,119)(117,127)(120,126)(124,128)Polycyclic presentation:
2^9|a^2=d,b^2=eg,c^2=ghi,d^2=f,f^2=h,h^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi,c^b=chi,d^b=degh,e^b=ei,g^b=gi,d^c=dg,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 938);