Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514000 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1970 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,17,21)(15,22,25,19)(16,26,18,20)(23,27,24,28);(2,3)(6,9)(10,12)(13,15,17,25)(14,19,21,22)(16,24,18,23)(20,28,26,27);(1,2,4,3)(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,20,21,26)(15,23,25,24)(19,27,22,28);(1,3,4,2)(14,21)(15,23)(16,18)(19,27)(22,28)(24,25);(6,9)(10,12)(13,16,17,18)(14,20,21,26)(15,24,25,23)(19,28,22,27)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7467682);