Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401030 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2541 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15123 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1817 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)(5,22)(6,12)(7,28)(8,13)(11,24)(14,40)(15,44)(16,31)(17,32)(18,47)(19,49)(20,36)(21,37)(23,39)(25,52)(26,33)(27,38)(29,42)(30,46)(34,43)(35,51)(41,62)(45,57)(48,55)(50,60)(53,56)(54,59)(58,63)(61,64);(1,3)(2,9)(4,18)(5,15)(6,26)(7,17)(8,16)(10,34)(11,30)(12,33)(13,31)(14,21)(19,48)(20,43)(22,44)(23,53)(24,46)(25,45)(27,42)(28,32)(29,37)(35,58)(36,47)(38,40)(39,56)(41,51)(49,61)(50,64)(52,57)(54,63)(55,60)(59,62);(1,4)(2,10)(3,14)(5,19)(6,27)(7,21)(8,20)(9,29)(11,35)(12,38)(13,36)(15,41)(16,42)(17,43)(18,26)(22,49)(23,54)(24,51)(25,50)(28,37)(30,55)(31,40)(32,47)(33,34)(39,59)(44,58)(45,63)(46,64)(48,53)(52,60)(56,61)(57,62);(1,5,7,24,8,25,6,23)(2,11,12,22,13,39,28,52)(3,15,17,46,16,45,26,53)(4,19,21,51,20,50,27,54)(9,30,33,44,31,56,32,57)(10,35,38,49,36,59,37,60)(14,41,43,64,42,63,18,48)(29,55,34,58,40,61,47,62)Polycyclic presentation:
2^9|d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,g^b=gi,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318825);