Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455690 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2825 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16340 |
| prime radical | #23412 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1111 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,12,8,14,7,13)(3,9,17,33,18,34,29,41)(4,19,22,37,23,40,28,10)(5,24,30,11,26,42,27,39)(15,43,45,35,20,52,46,31)(16,47,51,59,49,56,50,32)(21,36,55,60,25,38,54,53)(44,58,63,61,48,57,62,64);(1,3,6,17,8,18,7,29)(2,9,12,33,14,34,13,41)(4,20,22,46,23,15,28,45)(5,16,27,50,26,49,30,51)(10,35,19,52,37,31,40,43)(11,32,24,56,39,59,42,47)(21,48,54,63,25,44,55,62)(36,58,53,64,38,57,60,61);(1,4)(2,10)(3,15)(5,25)(6,28)(7,22)(8,23)(9,31)(11,38)(12,40)(13,19)(14,37)(16,48)(17,46)(18,20)(21,26)(24,53)(27,54)(29,45)(30,55)(32,58)(33,52)(34,35)(36,39)(41,43)(42,60)(44,49)(47,64)(50,62)(51,63)(56,61)(57,59);(1,5,6,27,8,26,7,30)(2,11,12,24,14,39,13,42)(3,16,17,50,18,49,29,51)(4,21,22,54,23,25,28,55)(9,32,33,56,34,59,41,47)(10,36,19,53,37,38,40,60)(15,44,45,62,20,48,46,63)(31,57,43,61,35,58,52,64)Polycyclic presentation:
2^9|a^2=ef,b^2=f,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366605);