Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417581 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1136 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)(6,9)(10,11)(13,14,17,19)(15,18,16,20)(21,22)(23,29)(24,27)(25,28)(26,31)(30,34)(32,36)(33,35);(5,6,8,9)(7,11,12,10)(13,15,17,16)(14,18,19,20)(21,23,25,32,24,31,33,30)(22,26,28,34,27,29,35,36);(13,16,17,15)(14,18,19,20)(22,27)(23,30)(25,33)(26,34)(29,36)(31,32);(2,4)(5,7,8,12)(6,10,9,11)(23,31)(26,29)(30,32)(34,36)Polycyclic presentation:
2^9|a^2=fi,b^2=efg,c^2=fi,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^c=gi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 346584);