Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
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 | |
1112 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)(5,6,8,10)(7,11,12,9)(13,14,17,19)(15,18,16,20)(21,22)(23,31)(24,29)(25,27)(26,30)(28,33)(32,35)(34,36);(5,7,8,12)(6,9,10,11)(13,15,17,16)(14,18,19,20)(22,27)(23,32)(24,26)(28,35)(31,36)(33,34);(13,16,17,15)(14,18,19,20)(21,23,25,33)(22,28,27,31)(24,32,26,34)(29,36,30,35);(2,4)(7,12)(9,11)(21,24,25,26)(22,29,27,30)(23,32,33,34)(28,36,31,35)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 333839);