Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2284 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,11)(7,12,8,10)(13,14)(15,21)(16,24)(17,19)(18,22)(20,26)(23,27)(25,28);(2,3)(5,6)(7,10)(8,12)(9,11)(14,19)(15,22)(20,24)(23,25);(1,2,4,3)(13,15,21,14,17,22,18,19)(16,25,28,24,26,23,27,20);(5,7,9,8)(6,10,11,12)(13,16)(14,20)(15,23)(17,26)(18,27)(19,24)(21,28)(22,25);(1,3,4,2)(5,8,9,7)(6,10,11,12)(16,26)(20,24)(23,25)(27,28)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6724231);