Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45332 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14)(15,23)(16,20)(17,30)(18,21)(19,25)(22,32)(24,39)(26,38)(27,41)(28,37)(29,44)(31,34)(33,40)(35,42)(36,43);(2,3)(5,6)(7,11)(8,10)(9,12);(5,7)(6,9)(8,12)(10,11)(13,15,18,26)(14,19,21,34)(16,27,28,24)(17,31,32,25)(20,35,37,33)(22,38,30,23)(29,40,43,42)(36,39,44,41);(1,3,4,2)(7,12)(9,11)(13,16,18,28)(14,20,21,37)(15,24,26,27)(17,29,32,43)(19,33,34,35)(22,36,30,44)(23,39,38,41)(25,40,31,42);(6,10)(9,11)(14,21)(16,28)(19,34)(22,30)(23,38)(24,27)(29,43)(40,42)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903745);