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 | |
2034 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)(16,27,28,36,30,34,24,20)(17,21,25,39,32,38,42,33)(29,43,41,37,31,44,40,35);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,26)(14,19,22,23)(16,28,30,24)(17,25,32,42)(20,34,36,27)(21,33,38,39)(29,41,31,40)(35,44,37,43);(5,7)(6,9)(8,12)(10,11)(13,16,18,30)(14,20,22,36)(15,24,26,28)(17,31,32,29)(19,27,23,34)(21,37,38,35)(25,41,42,40)(33,44,39,43);(1,3,4,2)(7,12)(9,11)(13,17)(14,21)(15,25)(16,29)(18,32)(19,33)(20,35)(22,38)(23,39)(24,40)(26,42)(27,43)(28,41)(30,31)(34,44)(36,37);(6,10)(9,11)(14,22)(17,32)(19,23)(20,36)(25,42)(27,34)(29,31)(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,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,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, 6913617);