Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| 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 | #38902 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(14,19)(15,22)(17,20)(21,26)(23,28)(25,27);(5,6,8,9)(7,10,11,12)(13,14,17,19,18,15,20,22)(16,23,25,28,24,21,27,26);(2,4)(6,9)(7,11)(13,15,17,22,18,14,20,19)(16,21,25,26,24,23,27,28);(5,7)(6,10)(8,11)(9,12)(13,16,18,24)(14,21,15,23)(17,25,20,27)(19,26,22,28);(6,9)(7,11)(16,24)(21,23)(25,27)(26,28)Polycyclic presentation:
2^9|b^2=fh,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^d=ehi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6505466);