Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161128 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#525 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
4200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,11,12,9)(8,10)(13,14)(15,23,25,28)(16,20)(17,22)(18,21)(19,24,27,26);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,24)(17,26)(18,25)(20,23)(21,27)(22,28);(2,4)(13,16,18,17)(14,20,21,22)(15,24,25,26)(19,23,27,28);(14,21)(19,27)(20,22)(23,28)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=ghi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72284499, #72284645, #72284678, … |
| maximal normal subgroup of | #72284499, #72284645, #72284678, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92233);