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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1895 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(13,14,19,23)(15,24,16,28)(17,21,20,27)(18,26,25,22);(5,6,8,9)(7,10,11,12)(13,15,25,21,19,16,18,27)(14,17,26,28,23,20,22,24);(2,4)(6,9)(7,11)(13,16,18,21,19,15,25,27)(14,20,22,28,23,17,26,24);(5,7)(6,10)(8,11)(9,12)(13,17,19,20)(14,21,23,27)(15,26,16,22)(18,24,25,28);(6,9)(7,11)(15,16)(17,20)(21,27)(24,28)Polycyclic presentation:
2^9|a^2=i,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,e^b=eh,d^c=dh,e^c=ei,e^d=eh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6505826);