Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393998 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14716 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2537 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)(3,15)(4,20)(5,11)(6,13)(7,12)(8,14)(9,18)(10,22)(16,29)(17,41)(19,37)(21,51)(23,38)(24,39)(25,36)(26,31)(27,34)(28,35)(30,46)(32,45)(33,53)(40,44)(42,58)(43,49)(47,57)(48,56)(50,54)(52,59)(55,62)(60,61)(63,64);(1,3,6,18,8,19,7,26)(2,9,12,15,14,31,13,37)(4,16,22,43,23,45,27,44)(5,17,24,46,25,48,28,47)(10,32,20,49,34,29,38,40)(11,30,35,41,36,57,39,56)(21,42,52,61,54,62,53,63)(33,58,50,64,59,55,51,60);(1,4)(2,10)(3,16)(5,21)(6,22)(7,27)(8,23)(9,29)(11,33)(12,20)(13,38)(14,34)(15,40)(17,42)(18,43)(19,45)(24,52)(25,54)(26,44)(28,53)(30,55)(31,32)(35,50)(36,59)(37,49)(39,51)(41,60)(46,61)(47,63)(48,62)(56,64)(57,58);(1,5)(2,11)(3,17)(4,21)(6,24)(7,28)(8,25)(9,30)(10,33)(12,35)(13,39)(14,36)(15,41)(16,42)(18,46)(19,48)(20,50)(22,52)(23,54)(26,47)(27,53)(29,55)(31,57)(32,58)(34,59)(37,56)(38,51)(40,60)(43,61)(44,63)(45,62)(49,64)Polycyclic presentation:
2^9|b^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,f^c=fi,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301402);