Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #244523 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3493 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
937 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 937 conjugacy classes of subgroups — too many to draw. Only its 144 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 96 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 96 points:
(1,2)(3,7)(4,8)(5,9)(6,10)(11,17)(12,18)(13,19)(14,27)(15,22)(16,23)(20,25)(21,28)(24,31)(26,30)(29,32)(33,34,40,45)(35,46,51,65)(36,52,57,69)(37,58,61,71)(38,62,39,44)(41,50,66,49)(42,55,70,56)(43,60,72,59)(47,75,77,63)(48,64,79,90)(53,73,81,67)(54,84,87,68)(74,80,89,78)(76,93,96,91)(82,86,92,85)(83,88,95,94);(1,3,6,13)(2,7,10,19)(4,11,15,25)(5,12,16,26)(8,20,22,17)(9,18,23,30)(14,24,28,32)(21,31,27,29)(33,35)(34,41)(36,53)(37,48)(38,50)(39,49)(40,51)(42,67)(43,64)(44,65)(45,66)(46,62)(47,55)(52,63)(54,83)(56,77)(57,81)(58,74)(59,80)(60,78)(61,79)(68,91)(69,75)(70,73)(71,89)(72,90)(76,86)(82,88)(84,93)(85,96)(87,95)(92,94);(1,4)(2,8)(3,11)(5,14)(6,15)(7,17)(9,21)(10,22)(12,24)(13,25)(16,28)(18,29)(19,20)(23,27)(26,32)(30,31)(33,36,40,57)(34,42,45,70)(35,47,51,77)(37,54,61,87)(38,55,39,56)(41,63,66,75)(43,68,72,84)(44,69,62,52)(46,73,65,67)(48,76,79,96)(49,53,50,81)(58,82,71,92)(59,85,60,86)(64,88,90,94)(74,93,89,91)(78,95,80,83);(1,5)(2,9)(3,12)(4,14)(6,16)(7,18)(8,21)(10,23)(11,24)(13,26)(15,28)(17,29)(19,30)(20,31)(22,27)(25,32)(33,37)(34,43)(35,48)(36,54)(38,60)(39,59)(40,61)(41,64)(42,68)(44,71)(45,72)(46,74)(47,76)(49,80)(50,78)(51,79)(52,82)(53,83)(55,86)(56,85)(57,87)(58,62)(63,88)(65,89)(66,90)(67,91)(69,92)(70,84)(73,93)(75,94)(77,96)(81,95)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,g^a=gi,c^b=cg,e^b=eh,g^b=gh,e^c=ei,f^c=fh,g^c=gh,e^d=eh,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 151737);