Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2711 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,18)(15,20,25,33)(16,21,27,34)(17,29,32,22)(23,35,39,26)(24,30,42,36)(28,38,31,37)(40,43,41,44);(5,7)(6,9)(8,12)(10,11)(13,15,19,25)(14,20,18,33)(16,26,27,35)(17,30,32,36)(21,23,34,39)(22,24,29,42)(28,40,31,41)(37,44,38,43);(2,4)(13,16)(14,21)(15,23)(17,31)(18,34)(19,27)(20,35)(22,38)(24,41)(25,39)(26,33)(28,32)(29,37)(30,44)(36,43)(40,42);(6,10)(9,11)(16,27)(21,34)(23,39)(26,35)(28,31)(37,38)(40,41)(43,44);(7,12)(9,11)(13,17,19,32)(14,22,18,29)(15,24,25,42)(16,28,27,31)(20,36,33,30)(21,37,34,38)(23,40,39,41)(26,43,35,44)Polycyclic presentation:
2^9|a^2=i,b^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,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6574843);