Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32696 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2749 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14539 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,12,6,29)(3,9,17,35,16,34,30,33)(4,18,23,37,21,47,22,10)(5,24,28,11,27,39,26,38)(14,40,42,31,19,48,50,56)(15,43,44,32,25,54,45,57)(20,36,53,59,51,60,52,46)(41,58,62,64,49,55,63,61);(1,3,6,17,8,16,7,30)(2,9,13,33,12,34,29,35)(4,19,22,50,21,14,23,42)(5,25,26,45,27,15,28,44)(10,31,18,40,37,56,47,48)(11,32,39,43,38,57,24,54)(20,49,52,63,51,41,53,62)(36,58,59,61,60,55,46,64);(1,4)(2,10,12,37)(3,14,16,19)(5,20)(6,22)(7,23)(8,21)(9,31)(11,36,38,60)(13,18,29,47)(15,41,25,49)(17,42,30,50)(24,46,39,59)(26,52)(27,51)(28,53)(32,55)(33,40)(34,56)(35,48)(43,64)(44,62,45,63)(54,61)(57,58);(1,5)(2,11)(3,15)(4,20)(6,28)(7,26)(8,27)(9,32)(10,36)(12,38)(13,24)(14,41)(16,25)(17,45)(18,46)(19,49)(21,51)(22,53)(23,52)(29,39)(30,44)(31,55)(33,54)(34,57)(35,43)(37,60)(40,61)(42,63)(47,59)(48,64)(50,62)(56,58)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fh,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^c=ei,g^c=gi,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296519);