Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| 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 | |
1081 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)(6,12)(7,17)(8,21)(9,10)(13,18)(14,29)(15,16)(19,33)(23,25)(26,28)(31,35)(32,36);(5,6,11,16)(7,18,17,26)(8,22,23,30)(9,15,10,12)(13,24,28,20)(14,25,29,21)(19,34,31,27)(32,35,36,33);(5,7,9,24,11,17,10,20)(6,13,12,26,16,28,15,18)(8,19,21,35,23,31,25,33)(14,27,22,36,29,34,30,32);(5,8,9,21,11,23,10,25)(6,14,12,22,16,29,15,30)(7,19,24,35,17,31,20,33)(13,27,26,36,28,34,18,32)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=ef,d^2=ef,e^2=i,g^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,g^b=gi,e^c=ei,f^c=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 292726);