Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,10)(7,11,8,12)(13,14,18,22)(15,24,26,19)(16,29,30,38)(17,21,33,23)(20,31,36,32)(25,40,42,44)(27,37,28,35)(34,41,39,43);(1,2)(3,4)(6,10)(11,12)(13,15,18,26)(14,19,22,24)(16,25,30,42)(17,28,33,27)(20,34,36,39)(21,37,23,35)(29,44,38,40)(31,43,32,41);(2,4)(5,7,9,8)(6,11,10,12)(13,16,17,32,18,30,33,31)(14,20,23,29,22,36,21,38)(15,25,28,41,26,42,27,43)(19,34,35,44,24,39,37,40);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(15,26)(16,30)(19,24)(20,36)(27,28)(29,38)(31,32)(35,37);(2,4)(5,8,9,7)(6,12,10,11)(13,17,18,33)(14,21,22,23)(15,27,26,28)(16,31,30,32)(19,35,24,37)(20,29,36,38)(25,41,42,43)(34,40,39,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,e^d=eh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6493001);