Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3036 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2924 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(9,10)(11,16)(12,17)(13,15)(14,18)(19,24)(20,22)(21,23);(6,7)(10,15)(11,19)(12,14)(16,23)(20,21)(22,24);(1,2,3,4)(5,6,8,7)(9,11,13,20)(10,16,15,22)(12,19,14,21)(17,24,18,23);(5,7,8,6)(9,11,13,20)(10,16,15,22)(12,19,14,21)(17,24,18,23);(9,12,13,14)(10,17,15,18)(11,19,20,21)(16,24,22,23)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,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 | #72312821, #72312922 |
| maximal normal subgroup of | #72312821, #72312922 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351367);