Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25196 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2424 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14204 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14,7,13,6,12)(3,15,20,9,19,34,18,33)(4,21,24,40,25,10,30,37)(5,26,28,11,29,38,27,39)(16,41,44,31,22,50,48,35)(17,32,47,42,46,58,45,57)(23,52,53,36,54,60,55,49)(43,59,62,64,51,56,63,61);(1,3,8,18,7,19,6,20)(2,9,14,15,13,33,12,34)(4,22,30,48,25,16,24,44)(5,17,27,47,29,46,28,45)(10,35,40,41,21,31,37,50)(11,32,26,42,39,58,38,57)(23,51,53,62,54,43,55,63)(36,59,60,61,49,56,52,64);(1,4)(2,10)(3,16)(5,23)(6,30)(7,25)(8,24)(9,31)(11,36)(12,40)(13,21)(14,37)(15,41)(17,43)(18,48)(19,22)(20,44)(26,52)(27,55)(28,53)(29,54)(32,56)(33,35)(34,50)(38,60)(39,49)(42,61)(45,63)(46,51)(47,62)(57,64)(58,59);(1,5,6,28,7,29,8,27)(2,11,12,38,13,39,14,26)(3,17,20,45,19,46,18,47)(4,23,30,53,25,54,24,55)(9,32,34,57,33,58,15,42)(10,36,40,60,21,49,37,52)(16,43,44,63,22,51,48,62)(31,56,50,64,35,59,41,61)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=fh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,e^d=ei,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285132);