Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14016 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,14,6,12)(3,9,17,30,18,31,26,37)(4,19,23,10,21,34,22,33)(5,11,24,35,25,36,27,38)(15,39,42,54,41,47,46,28)(16,29,45,57,44,56,43,55)(20,49,51,58,50,48,52,32)(40,60,63,53,61,59,62,64);(1,3,7,17,8,18,6,26)(2,9,13,30,14,31,12,37)(4,15,23,42,21,41,22,46)(5,16,24,45,25,44,27,43)(10,28,34,39,33,54,19,47)(11,29,35,57,36,56,38,55)(20,40,51,63,50,61,52,62)(32,53,49,59,58,64,48,60);(1,4)(2,10)(3,15,18,41)(5,20)(6,23)(7,22)(8,21)(9,28,31,54)(11,32)(12,34)(13,19)(14,33)(16,40,44,61)(17,46,26,42)(24,52)(25,50)(27,51)(29,53,56,64)(30,47,37,39)(35,48)(36,58)(38,49)(43,63,45,62)(55,59,57,60);(1,5)(2,11)(3,16)(4,20)(6,27)(7,24)(8,25)(9,29)(10,32)(12,38)(13,35)(14,36)(15,40)(17,45)(18,44)(19,48)(21,50)(22,52)(23,51)(26,43)(28,53)(30,57)(31,56)(33,58)(34,49)(37,55)(39,59)(41,61)(42,63)(46,62)(47,60)(54,64)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275431);