Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2149 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,8)(4,6)(5,7)(9,10)(11,15)(12,14)(13,16);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,18)(19,23)(20,22)(21,24);(10,14)(11,16);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,22)(19,20)(23,24);(2,7)(6,8)(10,14)(13,15)(17,19,21,20)(18,23,24,22)Polycyclic presentation:
2^9|b^2=f,d^2=fg,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,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72239653, #72256877, #72256922, … |
| maximal normal subgroup of | #72239653, #72256877, #72256922, … |
| maximal quotient of | #72239773, #72249797, #72264671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7455996);