Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395154 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×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 | |
1014 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)(21,22,26,32)(23,30,34,36)(24,29,27,31)(25,28,33,35);(6,9)(10,12)(13,14,16,18)(15,19,20,17)(21,23,24,25,26,34,27,33)(22,28,31,30,32,35,29,36);(2,4)(13,15,16,20)(14,17,18,19)(21,24,26,27)(22,29,32,31)(23,33,34,25)(28,30,35,36);(5,7,8,11)(6,10,9,12)(21,25,24,34,26,33,27,23)(22,30,31,35,32,36,29,28)Polycyclic presentation:
2^9|a^2=i,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^b=di,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318105);