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×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,11)(7,12,8,10)(13,14)(15,22)(16,26)(17,20)(18,21)(19,25)(23,27)(24,28);(5,6)(7,10)(8,12)(9,11)(13,15)(14,19)(16,24)(17,25)(18,23)(20,22)(21,28)(26,27);(1,2,3,4)(14,20)(15,23)(16,18)(19,27)(22,28)(24,25);(5,7,9,8)(6,10,11,12)(13,16,17,18)(14,21,20,26)(15,24,25,23)(19,28,22,27);(5,8,9,7)(6,10,11,12)(15,24)(16,18)(19,27)(21,26)(22,28)(23,25)Polycyclic presentation:
2^9|a^2=fi,c^2=gi,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6594253);