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 | |
2331 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,11)(8,12)(9,10)(13,14,16,18)(15,19,22,25)(17,20,23,26)(21,28,24,27);(2,3)(6,10)(11,12)(14,18)(15,21)(17,23)(19,27)(22,24)(25,28);(1,2,4,3)(5,7,9,8)(6,11,10,12)(15,22)(19,25)(21,24)(27,28);(1,3,4,2)(5,8,9,7)(6,12,10,11)(13,15)(14,19)(16,22)(17,24)(18,25)(20,27)(21,23)(26,28);(6,10)(11,12)(14,18)(15,22)(20,26)(21,24)Polycyclic presentation:
2^9|a^2=i,c^2=fgi,d^2=fgi,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, 7468761);