Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420424 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2699 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15278 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,31,7,13,6,12)(3,14,19,9,18,35,17,34)(4,20,25,38,24,10,23,39)(5,26,28,11,30,41,29,42)(15,44,49,36,21,43,46,32)(16,47,27,58,48,33,50,40)(22,53,55,37,56,63,57,51)(45,61,59,62,52,60,54,64);(1,3)(2,9)(4,21)(5,27)(6,19)(7,18)(8,17)(10,36)(11,40)(12,35)(13,34)(14,31)(15,24)(16,28)(20,32)(22,54)(23,46)(25,49)(26,47)(29,48)(30,50)(33,41)(37,62)(38,43)(39,44)(42,58)(45,57)(51,64)(52,55)(53,60)(56,59)(61,63);(1,4,6,23,7,24,8,25)(2,10,12,38,13,20,31,39)(3,15,17,46,18,21,19,49)(5,22,28,55,30,56,29,57)(9,32,14,43,34,36,35,44)(11,37,41,63,42,51,26,53)(16,45,27,59,48,52,50,54)(33,60,40,64,47,61,58,62);(1,5,6,28,7,30,8,29)(2,11,12,41,13,42,31,26)(3,16,17,27,18,48,19,50)(4,22,23,55,24,56,25,57)(9,33,14,40,34,47,35,58)(10,37,38,63,20,51,39,53)(15,45,46,59,21,52,49,54)(32,60,43,64,36,61,44,62)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^c=gi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327620);