Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 | |
1224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,11)(7,10,8,12)(13,14,16,18)(15,17,19,20)(21,22)(23,32)(24,29)(25,35)(26,36)(27,31)(28,33)(30,34);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,19)(18,20)(21,23,24,34,27,25,26,33)(22,28,36,35,31,30,29,32);(5,8,9,7)(6,10,11,12)(15,19)(17,20)(21,24,27,26)(22,29,31,36)(23,33,25,34)(28,35,30,32);(2,4)(21,25,27,23)(22,30,31,28)(24,34,26,33)(29,32,36,35)Polycyclic presentation:
2^9|a^2=fgi,b^2=efi,c^2=fi,d^2=i,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=de,e^d=ei,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342638);