Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,11)(8,9)(10,12)(13,14,17,19)(15,18,16,20)(21,22,25,30)(23,31,33,27)(24,28,26,29)(32,36,34,35);(1,2)(3,4)(6,9)(10,11)(13,15,17,16)(14,18,19,20)(21,23,25,33)(22,27,30,31)(24,32,26,34)(28,35,29,36);(2,4)(5,7)(6,10)(8,12)(9,11)(14,19)(18,20)(21,24,25,26)(22,28,30,29)(23,32,33,34)(27,35,31,36);(7,12)(10,11)(13,16,17,15)(14,20,19,18)(22,29)(24,26)(27,36)(28,30)(31,35)(32,34)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=i,d^2=efi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366915);