Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14526 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,30)(3,14,18,34,17,9,19,35)(4,20,23,10,24,38,31,39)(5,25,27,11,29,41,28,42)(15,36,21,32,45,59,46,43)(16,47,49,40,26,56,48,33)(22,37,54,62,55,63,53,50)(44,58,51,61,57,64,52,60);(1,3,6,19,8,17,7,18)(2,9,12,35,13,14,30,34)(4,21,23,45,24,46,31,15)(5,26,27,48,29,16,28,49)(10,36,20,43,39,59,38,32)(11,40,25,47,42,33,41,56)(22,52,53,57,55,51,54,44)(37,61,62,64,63,60,50,58);(1,4,7,23,8,24,6,31)(2,10,12,38,13,39,30,20)(3,15,18,21,17,45,19,46)(5,22,27,54,29,55,28,53)(9,32,35,59,14,43,34,36)(11,37,41,62,42,63,25,50)(16,44,49,51,26,57,48,52)(33,58,47,61,40,64,56,60);(1,5)(2,11)(3,16)(4,22)(6,28)(7,27)(8,29)(9,33)(10,37)(12,41)(13,42)(14,40)(15,44)(17,26)(18,49)(19,48)(20,50)(21,51)(23,54)(24,55)(25,30)(31,53)(32,58)(34,56)(35,47)(36,60)(38,62)(39,63)(43,64)(45,57)(46,52)(59,61)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,g^c=gi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295928);