Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| 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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3258 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)(13,14)(15,21)(16,20)(17,22)(18,19)(23,27)(24,26)(25,28);(5,6)(7,11)(8,10)(9,12)(13,15,21,14,18,22,17,19)(16,23,27,20,24,28,25,26);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(14,19)(15,22)(20,26)(23,28);(7,12)(9,11)(13,16)(14,20)(15,23)(17,25)(18,24)(19,26)(21,27)(22,28);(6,10)(9,11)(16,24)(20,26)(23,28)(25,27)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6413923);