Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 |
| prime radical | C2×#3309 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1891 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(13,14,17,22,18,19,26,28)(15,23,25,27,16,21,24,20);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(14,19)(15,16)(22,28)(24,25);(1,3,4,2)(6,9)(10,12)(13,15,18,16)(14,20,19,27)(17,24,26,25)(21,28,23,22);(5,7)(6,10)(8,11)(9,12)(13,16,18,15)(14,21,19,23)(17,24,26,25)(20,28,27,22);(7,11)(10,12)(13,17,18,26)(14,22,19,28)(15,24,16,25)(20,21,27,23)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6459736);