Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(25,26)(27,35)(28,33)(29,32)(30,34)(31,38)(36,47)(37,53)(39,45)(40,56)(41,48)(42,51)(43,49)(44,52)(46,55)(50,54);(9,11,14,21)(10,15,17,18)(12,19,13,20)(16,23,22,24)(25,27)(26,31)(28,40)(29,38)(30,36)(32,35)(33,50)(34,45)(37,44)(39,43)(41,54)(42,55)(46,52)(47,49)(48,56)(51,53);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(27,36)(30,43)(31,45)(34,49)(35,47)(37,54)(38,39)(40,55)(42,44)(46,56)(50,53)(51,52);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,32)(28,41)(31,35)(34,49)(37,55)(40,54)(42,44)(45,47);(11,21)(15,18)(19,20)(23,24)(25,28,29,41)(26,33,32,48)(27,37,38,55)(30,44,43,42)(31,46,35,53)(34,52,49,51)(36,54,39,40)(45,56,47,50)Polycyclic presentation:
2^9|a^2=fi,b^2=fi,c^2=fi,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6609695);