Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #425083 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2576 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15328 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
995 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,11,7,12,27,40)(3,13,16,30,17,38,25,8)(4,9,20,33,21,34,55,63)(5,10,26,39,24,37,23,36)(14,41,44,31,18,47,49,28)(15,35,46,29,45,42,22,56)(19,50,54,61,53,32,52,62)(43,60,51,59,48,64,57,58);(1,3)(2,8)(4,18)(5,22)(6,25)(7,17)(9,31)(10,35)(11,38)(12,30)(13,40)(14,21)(15,23)(16,27)(19,51)(20,44)(24,46)(26,45)(28,34)(29,36)(32,60)(33,41)(37,42)(39,56)(43,54)(47,63)(48,52)(49,55)(50,64)(53,57)(58,62)(59,61);(1,4,6,20,7,21,27,55)(2,9,11,33,12,34,40,63)(3,14,16,44,17,18,25,49)(5,19,26,54,24,53,23,52)(8,28,13,41,30,31,38,47)(10,32,39,62,37,50,36,61)(15,43,46,51,45,48,22,57)(29,58,42,60,56,59,35,64);(1,5)(2,10)(3,15)(4,19)(6,26)(7,24)(8,29)(9,32)(11,39)(12,37)(13,42)(14,43)(16,46)(17,45)(18,48)(20,54)(21,53)(22,25)(23,27)(28,58)(30,56)(31,59)(33,62)(34,50)(35,38)(36,40)(41,60)(44,51)(47,64)(49,57)(52,55)(61,63)Polycyclic presentation:
2^9|a^2=ef,c^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332279);