Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| 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 | #24391 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1900 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,24)(15,25,21,20)(16,27,28,26)(17,23,19,22);(1,2)(3,4)(6,9)(7,11)(13,15)(14,20)(16,23)(17,27)(18,21)(19,26)(22,28)(24,25);(2,4)(13,16,17,20,18,28,19,25)(14,21,22,26,24,15,23,27);(6,9)(10,12)(13,17,18,19)(14,22,24,23)(15,26,21,27)(16,25,28,20);(2,4)(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,23,24,22)(15,26,21,27)(16,20,28,25)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592469);