Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,15,23,18,22,26,19)(16,27,30,36,29,38,24,20)(17,31,25,34,33,21,32,37)(28,41,42,35,43,40,39,44);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,26)(14,19,22,23)(16,24,29,30)(17,32,33,25)(20,27,36,38)(21,37,31,34)(28,42,43,39)(35,41,44,40);(5,7)(6,9)(8,12)(10,11)(13,16,18,29)(14,20,22,36)(15,24,26,30)(17,28,33,43)(19,27,23,38)(21,35,31,44)(25,39,32,42)(34,40,37,41);(1,3,4,2)(7,12)(9,11)(13,17)(14,21)(15,25)(16,28)(18,33)(19,34)(20,35)(22,31)(23,37)(24,39)(26,32)(27,40)(29,43)(30,42)(36,44)(38,41);(6,10)(9,11)(17,33)(21,31)(25,32)(28,43)(34,37)(35,44)(39,42)(40,41)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 6913681);