Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394032 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14715 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3832 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)(4,20)(5,11)(6,29)(7,12)(8,13)(9,17)(10,24)(15,43)(16,46)(18,33)(19,32)(21,41)(22,51)(23,37)(25,36)(26,39)(27,38)(28,40)(30,45)(31,47)(34,50)(35,53)(42,48)(44,56)(49,57)(52,58)(54,59)(55,60)(61,63)(62,64);(1,3)(2,9)(4,21)(5,16)(6,18)(7,17)(8,19)(10,34)(11,31)(12,32)(13,33)(14,29)(15,23)(20,30)(22,52)(24,45)(25,50)(26,49)(27,47)(28,48)(35,58)(36,43)(37,41)(38,42)(39,46)(40,57)(44,54)(51,56)(53,64)(55,63)(59,61)(60,62);(1,4,8,25)(2,10,13,37)(3,15,19,45)(5,22,28,55)(6,23,7,24)(9,30,33,43)(11,35,40,59)(12,36,29,20)(14,41,32,34)(16,44,48,64)(17,21,18,50)(26,53,27,54)(31,56,57,62)(38,51,39,60)(42,61,46,58)(47,63,49,52);(1,5)(2,11)(3,16)(4,22)(6,27)(7,26)(8,28)(9,31)(10,35)(12,39)(13,40)(14,42)(15,44)(17,49)(18,47)(19,48)(20,51)(21,52)(23,54)(24,53)(25,55)(29,38)(30,56)(32,46)(33,57)(34,58)(36,60)(37,59)(41,61)(43,62)(45,64)(50,63)Polycyclic presentation:
2^9|c^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72306453, #72306459, #72306472, … |
| maximal normal subgroup of | #72306453, #72306459, #72306472, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301229);