Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885129 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1725 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(17,18,21,27,22,25,31,26)(19,28,30,24,20,23,29,32);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,20)(18,23,25,28)(21,30,31,29)(24,27,32,26);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,20,22,19)(18,24,25,32)(21,30,31,29)(23,27,28,26);(2,6)(5,7)(11,16)(13,15)(18,25)(19,20)(26,27)(29,30);(10,14)(13,15)(17,21,22,31)(18,26,25,27)(19,29,20,30)(23,32,28,24)Polycyclic presentation:
2^9|a^2=fgi,b^2=f,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72283519, #72283591, #72283690, … |
| maximal normal subgroup of | #72283519, #72283591, #72283690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6801434);