Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3179 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)(3,4)(13,14,19,23)(15,24,27,34)(16,21,29,37)(17,22,18,31)(20,32,35,26)(25,39,41,44)(28,36,30,38)(33,40,43,42);(5,6,8,9)(7,10,11,12)(13,15,19,27)(14,20,23,35)(16,25,29,41)(17,26,18,32)(21,33,37,43)(22,34,31,24)(28,40,30,42)(36,39,38,44);(2,4)(6,9)(7,11)(13,16,19,29)(14,21,23,37)(15,25,27,41)(17,30,18,28)(20,33,35,43)(22,38,31,36)(24,39,34,44)(26,42,32,40);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,28,29,30)(20,34,35,24)(21,36,37,38)(25,40,41,42)(33,39,43,44);(6,9)(7,11)(16,29)(21,37)(25,41)(28,30)(33,43)(36,38)(39,44)(40,42)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6502783);