Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3267 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2043 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,9)(7,11,12,10)(13,14)(15,24)(16,28)(17,32)(18,23)(19,22)(20,27)(21,31)(25,35)(26,36)(29,38)(30,39)(33,37)(34,40)(41,43)(42,44);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,29)(17,26)(18,27)(19,33)(21,38)(22,36)(23,24)(25,30)(28,44)(31,42)(32,37)(34,43)(35,40)(39,41);(1,2,3,4)(6,9)(10,11)(13,16)(14,21)(15,25)(17,30)(18,31)(19,34)(20,35)(22,39)(23,28)(24,41)(26,42)(27,43)(29,33)(32,40)(36,38)(37,44);(5,7,8,12)(6,10,9,11)(13,17,18,19)(14,22,23,32)(15,26,27,33)(16,30,31,34)(20,36,24,37)(21,39,28,40)(25,42,43,29)(35,38,41,44);(15,26)(17,19)(20,37)(22,32)(24,36)(25,42)(27,33)(29,43)(30,34)(35,44)(38,41)(39,40)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592208);