Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32888 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,19,22,18,23,16,21)(15,24,25,20,27,28,26,17);(1,2,3,4)(6,9)(10,11)(13,15)(14,20)(16,25)(17,23)(18,27)(19,26)(21,28)(22,24);(5,7,8,12)(6,10,9,11)(13,16,18,19)(14,21,23,22)(15,25,27,26)(17,24,20,28);(13,17,18,20)(14,15,23,27)(16,24,19,28)(21,25,22,26)Polycyclic presentation:
2^9|a^2=h,b^2=efh,c^2=fi,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,e^b=ei,g^b=gi,h^b=hi,e^d=ei,h^d=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300514);