Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460133 |
| 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 | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1154 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)(3,8,15,29)(4,9)(5,21)(6,24)(7,11)(10,23)(12,26)(13,37,17,27)(14,40,42,62)(16,45,44,30)(18,47)(19,32)(20,36)(22,50)(25,33)(28,55,56,43)(31,48)(34,52)(35,51)(38,60,41,63)(39,54,46,57)(49,59)(53,61,58,64);(1,3)(2,8,11,29)(4,17)(5,14)(6,16)(7,15)(9,27,32,37)(10,28,35,56)(12,30,24,45)(13,19)(18,41)(20,46)(21,40,50,62)(22,42)(23,43)(25,39)(26,44)(31,53,34,58)(33,54,36,57)(38,49)(47,63,59,60)(48,64)(51,55)(52,61);(1,4)(2,9)(3,13,15,17)(5,18)(6,25)(7,19)(8,27,29,37)(10,34)(11,32)(12,36)(14,41,42,38)(16,46,44,39)(20,26)(21,47)(22,49)(23,52)(24,33)(28,53,56,58)(30,57,45,54)(31,35)(40,60,62,63)(43,61,55,64)(48,51)(50,59);(1,5,7,22)(2,10)(3,14,15,42)(4,18,19,49)(6,23,26,51)(8,28)(9,31)(11,35)(12,21)(13,38,17,41)(16,43,44,55)(20,48,25,52)(24,50)(27,53)(29,56)(30,40)(32,34)(33,47)(36,59)(37,58)(39,61,46,64)(45,62)(54,63)(57,60)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=efi,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367330);