Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C4⋊C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,17,10,16,11,15)(7,18,20,12,21,13,26,28)(8,22,23,14,24,31,25,30)(19,33,34,32,35,27,36,29);(5,7,9,20,10,21,11,26)(6,12,15,18,16,28,17,13)(8,19,23,34,24,35,25,36)(14,27,22,32,30,33,31,29);(1,2,3,4)(5,7,10,21)(6,13,16,18)(8,19,24,35)(9,20,11,26)(12,17,28,15)(14,29,30,32)(22,27,31,33)(23,34,25,36);(5,8,9,23,10,24,11,25)(6,14,15,22,16,30,17,31)(7,19,20,34,21,35,26,36)(12,27,18,32,28,33,13,29)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267970);