Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,13)(12,14)(15,16);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(18,22)(19,20)(23,24);(1,3,5,4)(2,7,8,6)(10,14)(13,16)(17,18)(19,22)(20,24)(21,23);(1,4,5,3)(2,6,8,7)(17,18)(19,22)(20,24)(21,23);(17,19,21,20)(18,22,23,24)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72281985 |
| maximal normal subgroup of | #72281985 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349137);