Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460135 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16357 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1159 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)(3,8,15,30)(4,9)(5,21)(6,24)(7,11)(10,23)(12,27)(13,40,42,28)(14,43,45,29)(16,38,25,31)(17,32,49,59)(18,33)(19,52)(20,34)(22,37)(26,39)(35,54)(36,51)(41,63,44,55)(46,56,47,57)(48,58,64,62)(50,61)(53,60);(1,3)(2,8,11,30)(4,17)(5,14)(6,25)(7,15)(9,32,34,59)(10,29,21,43)(12,38,24,31)(13,19)(16,27)(18,48)(20,49)(22,47)(23,45)(26,46)(28,52,40,35)(33,58,36,62)(37,57,39,56)(41,53)(42,54)(44,50)(51,64)(55,60,63,61);(1,4)(2,9)(3,13,15,42)(5,18)(6,19)(7,20)(8,28,30,40)(10,36)(11,34)(12,35)(14,44,45,41)(16,17,25,49)(21,33)(22,50)(23,51)(24,52)(26,53)(27,54)(29,55,43,63)(31,32,38,59)(37,61)(39,60)(46,48,47,64)(56,62,57,58);(1,5,7,23)(2,10)(3,14,15,45)(4,18,20,51)(6,26,27,22)(8,29)(9,33)(11,21)(12,39)(13,41,42,44)(16,47,25,46)(17,48,49,64)(19,53,54,50)(24,37)(28,55)(30,43)(31,57)(32,58)(34,36)(35,61)(38,56)(40,63)(52,60)(59,62)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=efi,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72285925, #72286106, #72286154, … |
| maximal normal subgroup of | #72285925, #72286106, #72286154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367332);