Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996719 |
| 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 | #45245 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2020 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)(5,6)(7,9)(8,10)(11,12)(13,14,17,22,18,23,31,37)(15,24,27,36,26,20,34,19)(16,28,32,42,33,21,39,41)(25,38,29,35,30,40,43,44);(5,6)(7,11)(8,10)(9,12)(13,15,18,26)(14,19,23,36)(16,29,33,43)(17,34,31,27)(20,37,24,22)(21,40,28,38)(25,39,30,32)(35,42,44,41);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,30)(17,27)(18,26)(19,22)(21,40)(23,24)(25,33)(28,38)(29,39)(31,34)(32,43)(35,42)(36,37)(41,44);(7,12)(9,11)(13,16,31,39,18,33,17,32)(14,21,37,42,23,28,22,41)(15,25,34,43,26,30,27,29)(19,35,20,38,36,44,24,40);(6,10)(9,11)(15,26)(19,36)(20,24)(25,30)(27,34)(29,43)(35,44)(38,40)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,d^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6914349);