Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907909 |
| 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 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1619 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,8)(5,7)(9,10,12,14)(11,15,16,13)(19,25)(20,27)(22,28)(24,30)(26,31)(29,32);(1,2,6,3)(4,7,5,8)(9,11,12,16)(10,13,14,15)(17,18,21,23)(19,22,26,29)(20,24,27,30)(25,28,31,32);(1,3,6,2)(4,7,5,8)(11,16)(13,15)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(18,23)(19,26)(24,30)(25,31);(1,5,6,4)(2,7,3,8)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=g,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfhi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72340737, #72340740 |
| maximal normal subgroup of | #72340737, #72340740 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6897329);