Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32870 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2826 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14621 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,14,19,43)(4,20,25,10)(5,11)(6,12)(7,31)(8,13)(9,17,34,18)(15,35)(16,38,50,60)(21,32)(22,52,54,36)(23,44,24,37)(26,33,51,61)(27,56,55,39)(28,40)(29,41)(30,42)(45,48)(46,62)(47,58)(49,57)(53,59)(63,64);(1,3)(2,9)(4,21)(5,26)(6,18)(7,17)(8,19)(10,35)(11,38)(12,43)(13,34)(14,31)(15,24)(16,28)(20,45)(22,53)(23,48)(25,47)(27,49)(29,51)(30,50)(32,37)(33,42)(36,63)(39,59)(40,61)(41,60)(44,58)(46,54)(52,57)(55,64)(56,62);(1,4)(2,10)(3,15)(5,27)(6,24)(7,23)(8,25)(9,32)(11,39)(12,44)(13,20)(14,45)(16,49)(17,21)(18,47)(19,48)(22,28)(26,46)(29,55)(30,54)(31,37)(33,59)(34,58)(35,43)(36,40)(38,63)(41,56)(42,52)(50,64)(51,53)(57,60)(61,62);(1,5)(2,11)(3,16)(4,22)(6,30)(7,28)(8,29)(9,33)(10,36)(12,42)(13,41)(14,38)(15,46)(17,51)(18,26)(19,50)(20,52)(21,49)(23,55)(24,27)(25,54)(31,40)(32,57)(34,61)(35,62)(37,56)(39,44)(43,60)(45,59)(47,64)(48,53)(58,63)Polycyclic presentation:
2^9|a^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,d^b=def,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298403);