Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,11,15)(7,16,19,30)(8,13,24,29)(9,21,14,10)(12,25,28,18)(17,32,20,35)(22,23,31,26)(27,33,36,34);(5,7,11,19)(6,12,15,28)(8,20,24,17)(9,16,14,30)(10,25,21,18)(13,27,29,36)(22,32,31,35)(23,33,26,34);(5,8,21,23,11,24,10,26)(6,13,9,22,15,29,14,31)(7,17,25,33,19,20,18,34)(12,27,30,35,28,36,16,32);(5,9,21,15,11,14,10,6)(7,12,25,30,19,28,18,16)(8,22,23,29,24,31,26,13)(17,27,33,35,20,36,34,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=ef,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292723);