Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342836 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C2×#3492 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4⋊C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,23,27,30)(21,24)(25,28,31,32)(26,29);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,21,26)(18,22,24,29)(20,25,27,31)(23,28,30,32);(17,20)(18,23,24,30)(19,25)(21,27)(22,28,29,32)(26,31);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15);(2,8)(6,9)(7,16)(14,15)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=gh,d^2=f,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,g^c=gi,h^c=hi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72310548, #72310549, #72310550, … |
| maximal normal subgroup of | #72310548, #72310549, #72310550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300877);