Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| 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 | |
882 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)(13,14,16,18)(15,17,19,20)(21,22,26,32)(23,28,27,31)(24,30,33,35)(25,29,34,36);(6,9)(10,12)(13,15,16,19)(14,17,18,20)(21,23,26,27)(22,28,32,31)(24,34,33,25)(29,30,36,35);(5,7)(6,10)(8,11)(9,12)(15,19)(17,20)(21,24,27,34,26,33,23,25)(22,29,28,35,32,36,31,30);(2,4)(7,11)(10,12)(21,25,23,33,26,34,27,24)(22,30,31,36,32,35,28,29)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fi,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=di,d^c=de68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 375085);