Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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) = 128 |
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)(15,20,26,25)(16,21,29,32)(17,22,30,31)(23,33,37,27)(24,34,38,39)(28,35,41,42)(36,43,44,40);(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,27)(17,24)(18,25)(19,26)(21,23)(22,34)(28,40)(29,33)(30,38)(31,39)(32,37)(35,36)(41,43)(42,44);(2,4)(13,16,19,29)(14,21,18,32)(15,23,26,37)(17,28,30,41)(20,33,25,27)(22,35,31,42)(24,36,38,44)(34,43,39,40);(6,10)(9,11)(13,17)(14,22)(15,24)(16,28)(18,31)(19,30)(20,34)(21,35)(23,36)(25,39)(26,38)(27,40)(29,41)(32,42)(33,43)(37,44);(7,12)(9,11)(14,18)(15,25)(17,30)(20,26)(21,32)(23,27)(24,34)(28,41)(33,37)(36,43)(38,39)(40,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^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, 6575668);