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 | #36018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2012 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)(16,22)(17,23)(18,34)(19,25)(20,26)(24,27)(28,37)(29,39)(30,40)(31,41)(32,38)(33,43)(35,44)(36,42);(5,7,8,12)(6,9,10,11)(13,15,20,29)(14,21,26,39)(16,30,32,33)(17,28,19,35)(18,31,27,36)(22,40,38,43)(23,37,25,44)(24,42,34,41);(1,2,3,4)(13,16)(14,22)(15,27)(17,31)(18,29)(19,36)(20,32)(21,24)(23,41)(25,42)(26,38)(28,33)(30,35)(34,39)(37,43)(40,44);(6,10)(9,11)(13,17,20,19)(14,23,26,25)(15,28,29,35)(16,31,32,36)(18,30,27,33)(21,37,39,44)(22,41,38,42)(24,43,34,40);(7,12)(9,11)(13,18,20,27)(14,24,26,34)(15,16,29,32)(17,33,19,30)(21,38,39,22)(23,40,25,43)(28,36,35,31)(37,41,44,42)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6590108);