Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1983 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)(7,11)(8,10)(9,12)(13,14)(15,23)(16,24)(17,28)(18,22)(19,27)(20,25)(21,26);(2,3)(5,7)(6,9)(8,12)(10,11)(13,15,18,25)(14,20,22,23)(16,21,27,28)(17,19,26,24);(1,2,4,3)(13,16,24,14,18,27,19,22)(15,21,17,20,25,28,26,23);(6,10)(9,11)(13,17,18,26)(14,21,22,28)(15,24,25,19)(16,23,27,20);(1,3,4,2)(7,12)(9,11)(15,25)(17,26)(20,23)(21,28)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,d^2=i,e^2=h,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=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6721849);