Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427936 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15376 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1238 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)(3,9)(4,10)(5,24,26,40)(6,28)(7,12)(8,13)(11,27,39,29)(14,43)(15,47,25,57)(16,32)(17,41)(18,33)(19,34)(20,51,53,56)(21,42)(22,36)(23,37)(30,45)(31,48,38,49)(35,54,63,55)(44,61,58,62)(46,60)(50,59,52,64);(1,3)(2,9)(4,19)(5,25)(6,18)(7,17)(8,16)(10,34)(11,38)(12,41)(13,32)(14,21)(15,26)(20,52)(22,45)(23,46)(24,47)(27,48)(28,33)(29,49)(30,36)(31,39)(35,62)(37,60)(40,57)(42,43)(44,55)(50,53)(51,59)(54,58)(56,64)(61,63);(1,4,8,23)(2,10,13,37)(3,14,16,45)(5,20,26,53)(6,22,7,21)(9,30,32,43)(11,35,39,63)(12,42,28,36)(15,44,25,58)(17,46,18,19)(24,56,40,51)(27,55,29,54)(31,59,38,64)(33,60,41,34)(47,61,57,62)(48,52,49,50);(1,5)(2,11)(3,15,16,25)(4,20)(6,27)(7,29)(8,26)(9,31,32,38)(10,35)(12,40)(13,39)(14,44,45,58)(17,48,18,49)(19,50,46,52)(21,54)(22,55)(23,53)(24,28)(30,59,43,64)(33,57,41,47)(34,61,60,62)(36,56)(37,63)(42,51)Polycyclic presentation:
2^9|a^2=e,c^2=i,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335133);