Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682420 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36124 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,11,12,10)(13,14)(15,19)(16,23)(17,26)(18,21)(20,25)(22,27)(24,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,17)(14,19,21,26)(16,22,24,25)(20,23,27,28);(2,4)(6,9)(10,11)(13,16,17,25,18,24,15,22)(14,20,19,23,21,27,26,28);(5,7,8,12)(6,10,9,11)(16,24)(20,27)(22,25)(23,28);(2,4)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308225, #72308371, #72308554 |
| maximal normal subgroup of | #72308225, #72308371, #72308554 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592038);