Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | #30502 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1851 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(6,9)(7,11)(13,14,18,24)(15,25,28,20)(16,29,31,22)(17,23,19,34)(21,33,39,30)(26,36,43,41)(27,38,32,37)(35,44,40,42);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15,18,28)(14,20,24,25)(16,26,31,43)(17,32,19,27)(21,35,39,40)(22,36,29,41)(23,38,34,37)(30,44,33,42);(1,2,4,3)(13,16)(14,21)(15,26)(17,30)(18,31)(19,33)(20,35)(22,23)(24,39)(25,40)(27,42)(28,43)(29,34)(32,44)(36,38)(37,41);(5,7,8,11)(6,10,9,12)(13,16)(14,22)(15,26)(17,33)(18,31)(19,30)(20,36)(21,23)(24,29)(25,41)(27,44)(28,43)(32,42)(34,39)(35,38)(37,40);(1,3,4,2)(6,9)(10,12)(13,17,18,19)(14,23,24,34)(15,27,28,32)(16,30,31,33)(20,37,25,38)(21,22,39,29)(26,42,43,44)(35,41,40,36)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=h,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,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, 6715486);