Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667676 |
| 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 |
| prime radical | #36123 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1847 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(10,13)(11,15)(14,16)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(1,2)(3,7)(4,5)(6,8)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(2,5)(6,7)(9,10,12,14)(11,16,15,13)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31);(18,24)(22,25)(23,30)(29,31);(1,3,4,8)(2,6,5,7)(10,14)(13,16)(19,27)(22,25)(26,32)(29,31)Polycyclic presentation:
2^9|c^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72283055, #72283164, #72283241, … |
| maximal normal subgroup of | #72283055, #72283164, #72283241, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6698682);