Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33688 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #3109 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #21122 |
| prime radical | #3414 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,15,6,13,9,33)(3,16,21,48,19,10,22,38)(4,23,25,55,26,39,34,46)(5,14,7,30,28,44,29,12)(11,32,40,57,42,58,41,24)(17,49,51,63,52,60,54,64)(18,45,20,36,27,37,31,43)(35,59,61,50,47,53,62,56);(1,3,8,21,6,19,9,22)(2,10,15,38,13,16,33,48)(4,17,25,51,26,52,34,54)(5,27,7,31,28,18,29,20)(11,35,40,61,42,47,41,62)(12,43,14,45,30,36,44,37)(23,49,55,63,39,60,46,64)(24,56,32,59,57,50,58,53);(1,4)(2,11)(3,17)(5,24)(6,26)(7,32)(8,25)(9,34)(10,35)(12,39)(13,42)(14,46)(15,40)(16,47)(18,50)(19,52)(20,53)(21,51)(22,54)(23,30)(27,56)(28,57)(29,58)(31,59)(33,41)(36,49)(37,63)(38,61)(43,60)(44,55)(45,64)(48,62);(1,5,8,7,6,28,9,29)(2,12,33,44,13,30,15,14)(3,18,21,20,19,27,22,31)(4,24,25,32,26,57,34,58)(10,36,48,45,16,43,38,37)(11,39,41,55,42,23,40,46)(17,50,51,53,52,56,54,59)(35,49,62,64,47,60,61,63)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=eh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,h^a=hi,d^b=di,f^b=fi,e^c=ei,f^c=fh,h^c=hi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402196);