Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #23848 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,25,23,26,21,20)(19,27,30,31,28,24,32,29);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,20)(19,28)(21,22)(24,29)(25,26)(27,31);(2,6)(5,7)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,31)(21,32)(22,30)(23,28)(25,29)(26,27);(3,8)(5,7)(11,16)(13,15)(17,20,22,18,23,25,21,26)(19,29,30,27,28,31,32,24);(10,14)(13,15)(17,21,23,22)(18,25,26,20)(19,30,28,32)(24,31,27,29)Polycyclic presentation:
2^9|a^2=fgi,c^2=hi,d^2=fgi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308949, #72308967 |
| maximal normal subgroup of | #72308949, #72308967 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6987954);