Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2018 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)(5,6,8,10)(7,9,11,12)(13,14)(15,23)(16,28)(17,21)(18,22)(19,26)(20,30)(24,35)(25,34)(27,33)(29,36)(31,41)(32,38)(37,42)(39,43)(40,44);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,29)(17,26)(18,24)(20,36)(21,23)(22,33)(25,30)(27,32)(28,34)(31,40)(35,38)(37,44)(39,42)(41,43);(2,4)(5,7,8,11)(6,9,10,12)(15,24)(18,32)(19,33)(22,38)(23,35)(25,39)(26,27)(29,40)(31,42)(34,43)(36,44)(37,41);(2,4)(6,10)(9,12)(13,16)(14,20)(15,25)(17,30)(18,31)(19,34)(21,28)(22,37)(23,36)(24,39)(26,29)(27,40)(32,42)(33,43)(35,44)(38,41);(14,21)(15,26)(16,30)(22,38)(24,27)(31,42)(34,36)(43,44)Polycyclic presentation:
2^9|a^2=fi,c^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408649);