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 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1888 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)(7,10)(8,9)(11,12)(13,14,18,19)(15,22,25,20)(16,26,17,21)(23,27,28,24);(6,9)(7,11)(13,15,18,25)(14,20,19,22)(16,24,17,27)(21,23,26,28);(1,2,3,4)(13,16,18,17)(14,21,19,26)(15,23,25,28)(20,27,22,24);(6,9)(10,12)(13,17,18,16)(14,21,19,26)(15,24,25,27)(20,28,22,23);(5,7,8,11)(6,10,9,12)(14,19)(15,20)(21,26)(22,25)(23,27)(24,28)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592365);