Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394133 |
| 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 | |
3840 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)(4,10)(5,24)(6,13)(7,12)(8,14)(9,18)(11,25)(16,42)(17,31)(19,33)(20,30)(21,50)(22,37)(23,38)(26,39)(27,32)(28,36)(29,40)(34,44)(35,52)(41,48)(43,62)(45,56)(46,54)(47,57)(49,61)(51,60)(53,59)(55,63)(58,64);(1,3)(2,9)(4,20)(5,17)(6,27)(7,18)(8,19)(10,34)(11,31)(12,33)(13,15)(14,32)(16,22)(21,49)(23,45)(24,54)(25,48)(26,47)(28,44)(29,46)(30,37)(35,58)(36,56)(38,42)(39,41)(40,57)(43,51)(50,55)(52,63)(53,64)(59,62)(60,61);(1,4)(2,10)(3,16)(5,21)(6,28)(7,22)(8,23)(9,30)(11,35)(12,37)(13,36)(14,38)(15,34)(17,43)(18,45)(19,44)(20,27)(24,50)(25,52)(26,53)(29,51)(31,55)(32,56)(33,42)(39,59)(40,60)(41,58)(46,64)(47,63)(48,49)(54,61)(57,62);(1,5,8,26)(2,11,14,40)(3,17,19,47)(4,21,23,53)(6,25,7,29)(9,31,32,57)(10,35,38,60)(12,24,13,39)(15,41,33,54)(16,43,44,63)(18,46,27,48)(20,49,45,64)(22,51,28,52)(30,55,56,62)(34,58,42,61)(36,59,37,50)Polycyclic presentation:
2^9|d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72286957, #72287270, #72287286, … |
| maximal normal subgroup of | #72286957, #72287270, #72287286, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301330);