Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,3,4)(9,10)(11,19)(12,23)(13,26)(14,17)(15,22)(16,24)(18,29)(20,30)(21,33)(25,34)(27,31)(28,32)(35,37)(36,40)(38,39);(1,2)(3,4)(5,6)(7,8);(6,8)(9,11)(10,15)(12,20)(13,27)(14,21)(16,30)(17,31)(18,33)(19,26)(22,29)(23,35)(24,38)(25,36)(28,37)(32,40)(34,39);(2,4)(9,12,14,25)(10,16,17,32)(11,20,21,36)(13,28,29,24)(15,30,31,40)(18,34,26,23)(19,35,33,39)(22,38,27,37);(10,17)(11,21)(16,32)(18,26)(20,36)(22,27)(23,34)(37,38)Polycyclic presentation:
2^9|a^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408167);