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 | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
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)(23,33)(24,31)(25,28)(26,29)(27,32)(30,34)(35,36);(6,9)(10,12)(13,14,16,18)(15,20,19,17)(21,23,24,35,27,34,26,25)(22,28,29,30,32,36,31,33);(2,4)(13,15,16,19)(14,17,18,20)(21,24,27,26)(22,29,32,31)(23,25,34,35)(28,33,36,30);(5,7,8,11)(6,10,9,12)(21,25,26,34,27,35,24,23)(22,30,31,28,32,33,29,36)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,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^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318363);