Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45092 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1663 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,22,24)(19,28,31,40)(20,26,32,38)(21,27,23,36)(25,34,37,30)(29,43,44,48)(33,41,35,42)(39,45,47,46);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(18,24)(20,32)(25,37)(27,36)(28,40)(29,44)(33,35)(45,46);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,19,21,34,22,31,23,30)(18,25,36,40,24,37,27,28)(20,29,35,45,32,44,33,46)(26,39,41,48,38,47,42,43);(3,8)(6,7)(11,16)(13,15)(17,20)(18,26)(19,29)(21,35)(22,32)(23,33)(24,38)(25,39)(27,42)(28,43)(30,46)(31,44)(34,45)(36,41)(37,47)(40,48);(10,14)(13,15)(17,21,22,23)(18,27,24,36)(19,30,31,34)(20,33,32,35)(25,40,37,28)(26,41,38,42)(29,45,44,46)(39,43,47,48)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=fg,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6927628);