Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3393 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,19)(16,20)(17,22)(18,21)(23,26)(24,27)(25,28);(5,7)(6,9)(8,12)(10,11)(13,15)(14,19)(16,24)(17,23)(18,25)(20,27)(21,28)(22,26);(2,4)(15,23)(16,17)(19,26)(20,22)(24,25)(27,28);(6,10)(9,11)(13,16,18,17)(14,20,21,22)(15,24,25,23)(19,27,28,26);(7,12)(9,11)(14,21)(15,23)(16,17)(19,27)(24,25)(26,28)Polycyclic presentation:
2^9|d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281973, #72282845, #72283091, … |
| maximal normal subgroup of | #72281973, #72282845, #72283091, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6575066);