Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2056 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14,18,24)(15,25,26,20)(16,30,33,22)(17,23,19,21)(27,36,31,39)(28,37,29,35)(32,40,34,38)(41,44,42,43);(2,3)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,31)(17,28)(18,26)(19,29)(21,37)(22,39)(23,35)(24,25)(27,33)(30,36)(32,42)(34,41)(38,43)(40,44);(1,2,4,3)(14,21)(15,26)(17,19)(20,35)(22,38)(23,24)(25,37)(27,31)(30,40)(32,34)(36,43)(39,44);(6,10)(9,11)(13,16,18,33)(14,22,24,30)(15,27,26,31)(17,34,19,32)(20,36,25,39)(21,38,23,40)(28,42,29,41)(35,43,37,44);(1,3,4,2)(7,12)(9,11)(13,17,18,19)(14,23,24,21)(15,28,26,29)(16,32,33,34)(20,35,25,37)(22,38,30,40)(27,41,31,42)(36,44,39,43)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6714361);