Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,21)(16,30)(17,23)(18,24)(19,25)(20,26)(22,28)(27,38)(29,39)(31,42)(32,44)(33,41)(34,40)(35,37)(36,43);(5,7,8,12)(6,9,10,11)(13,15,19,29)(14,21,25,39)(16,31,28,34)(17,27,20,36)(18,33,35,32)(22,40,30,42)(23,38,26,43)(24,41,37,44);(1,2,3,4)(13,16)(14,22)(15,18)(17,32)(19,28)(20,33)(21,37)(23,41)(24,39)(25,30)(26,44)(27,34)(29,35)(31,36)(38,42)(40,43);(6,10)(9,11)(13,17,19,20)(14,23,25,26)(15,27,29,36)(16,32,28,33)(18,34,35,31)(21,38,39,43)(22,41,30,44)(24,40,37,42);(7,12)(9,11)(13,18,19,35)(14,24,25,37)(15,28,29,16)(17,31,20,34)(21,22,39,30)(23,42,26,40)(27,32,36,33)(38,44,43,41)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=i,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, 6590030);