Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500689 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,9)(7,11,12,10)(13,14)(15,19)(16,27)(17,21)(18,23)(20,29)(22,31)(24,38)(25,33)(26,35)(28,37)(30,36)(32,40)(34,42)(39,44)(41,43);(1,2)(3,4)(6,9)(7,12)(13,15,18,26)(14,19,23,35)(16,24,28,39)(17,25,31,42)(20,32,36,43)(21,33,22,34)(27,38,37,44)(29,40,30,41);(2,4)(13,16)(14,20)(15,24)(17,30)(18,28)(19,32)(21,27)(22,37)(23,36)(25,41)(26,39)(29,31)(33,38)(34,44)(35,43)(40,42);(5,7,8,12)(6,10,9,11)(14,21)(17,31)(19,33)(20,27)(22,23)(25,42)(29,30)(32,38)(34,35)(36,37)(40,41)(43,44);(16,28)(20,36)(24,39)(27,37)(29,30)(32,43)(38,44)(40,41)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,e^c=ei,g^c=gi,g^d=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409517);