Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3427 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)(7,10)(8,9)(11,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,23)(24,27)(25,28);(1,2)(3,4)(13,15,17,23,18,24,16,25)(14,19,20,22,21,28,26,27);(2,4)(6,9)(10,12);(2,4)(13,16,18,17)(14,20,21,26)(15,23,24,25)(19,27,28,22);(5,7,8,11)(6,10,9,12)(15,24)(19,28)(22,27)(23,25)Polycyclic presentation:
2^9|b^2=f,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^b=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72263943, #72263954, #72264047, … |
| maximal normal subgroup of | #72263943, #72263954, #72264047, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361766);