Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682532 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(7,11)(13,14,17,21)(15,23,20,19)(16,25,27,24)(18,28,26,22);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,22)(17,20)(18,24)(21,23)(25,26)(27,28);(1,2,4,3)(6,9)(10,12)(13,16,26,19,17,27,18,23)(14,20,28,24,21,15,22,25);(5,7,8,11)(6,10,9,12)(14,21)(15,20)(22,28)(24,25);(1,3,4,2)(15,20)(16,27)(19,23)(24,25)Polycyclic presentation:
2^9|a^2=i,c^2=fgh,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724398);