Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(14,19)(16,23)(20,25)(22,24);(5,7,8,12)(6,9,10,11)(14,20)(15,18)(16,22)(19,25)(23,24)(27,28);(1,2,3,4)(13,14)(15,20)(16,21)(17,19)(18,25)(22,27)(23,26)(24,28);(6,10)(9,11)(13,15,17,18)(14,20,19,25)(16,24,23,22)(21,28,26,27);(7,12)(9,11)(13,16)(14,21)(15,22)(17,23)(18,24)(19,26)(20,27)(25,28)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590140);