Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395154 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1076 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(15,19)(17,20)(21,22)(23,30)(24,28)(25,32)(26,29)(27,31)(33,36)(34,35);(6,9)(10,12)(13,14,16,18)(15,20,19,17)(21,23,25,24,26,33,27,34)(22,28,31,30,29,35,32,36);(2,4)(13,15,16,19)(14,17,18,20)(22,29)(28,35)(30,36)(31,32);(5,7,8,11)(6,10,9,12)(21,24,25,33,26,34,27,23)(22,30,31,35,29,36,32,28)Polycyclic presentation:
2^9|a^2=fgi,b^2=ghi,c^2=g,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318368);