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)(13,14,20,26)(15,21,30,39)(16,22,29,38)(17,23,19,25)(18,24,27,36)(28,37,35,44)(31,40,33,42)(32,41,34,43);(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,31)(17,28)(18,34)(19,35)(20,30)(22,40)(23,37)(24,43)(25,44)(26,39)(27,32)(29,33)(36,41)(38,42);(1,2,3,4)(13,16,20,29)(14,22,26,38)(15,27,30,18)(17,32,19,34)(21,36,39,24)(23,41,25,43)(28,33,35,31)(37,42,44,40);(6,10)(9,11)(13,17,20,19)(14,23,26,25)(15,28,30,35)(16,32,29,34)(18,31,27,33)(21,37,39,44)(22,41,38,43)(24,40,36,42);(7,12)(9,11)(13,18,20,27)(14,24,26,36)(15,29,30,16)(17,33,19,31)(21,38,39,22)(23,42,25,40)(28,32,35,34)(37,41,44,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,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, 6590029);