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 |
| prime radical | #45136 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2019 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)(16,27,28,37,30,35,24,20)(17,31,25,34,33,21,38,40)(29,44,42,36,32,43,41,39);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,18,26)(14,19,22,23)(16,28,30,24)(17,25,33,38)(20,35,37,27)(21,34,31,40)(29,42,32,41)(36,44,39,43);(5,7)(6,9)(8,12)(10,11)(13,16,18,30)(14,20,22,37)(15,24,26,28)(17,32,33,29)(19,27,23,35)(21,39,31,36)(25,42,38,41)(34,44,40,43);(1,3,4,2)(7,12)(9,11)(13,17,18,33)(14,21,22,31)(15,25,26,38)(16,29,30,32)(19,34,23,40)(20,36,37,39)(24,41,28,42)(27,43,35,44);(6,10)(9,11)(14,22)(17,33)(19,23)(20,37)(25,38)(27,35)(29,32)(41,42)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6913871);