Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,9)(7,11,12,10)(13,14,18,24)(15,26,30,20)(16,31,33,22)(17,25,19,23)(21,35,39,32)(27,40,28,37)(29,38,34,41)(36,44,42,43);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,30)(14,20,24,26)(16,27,33,28)(17,34,19,29)(21,36,39,42)(22,40,31,37)(23,41,25,38)(32,44,35,43);(2,4)(6,9)(10,11)(13,16)(14,21)(15,27)(17,32)(18,33)(19,35)(20,36)(22,25)(23,31)(24,39)(26,42)(28,30)(29,43)(34,44)(37,41)(38,40);(5,7,8,12)(6,10,9,11)(13,16)(14,22)(15,28)(17,35)(18,33)(19,32)(20,37)(21,25)(23,39)(24,31)(26,40)(27,30)(29,43)(34,44)(36,41)(38,42);(2,4)(13,17,18,19)(14,23,24,25)(15,29,30,34)(16,32,33,35)(20,38,26,41)(21,31,39,22)(27,43,28,44)(36,40,42,37)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6578507);