Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32898 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,8,5,7,11,14)(3,6,9,13,10,12,15,16)(17,18,22,26,23,25,21,19)(20,28,29,32,30,24,31,27);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,19,21,25,23,26,22,18)(20,27,29,28,30,32,31,24);(17,20)(18,24)(19,27)(21,29)(22,31)(23,30)(25,28)(26,32);(2,7)(6,12)(8,14)(13,16)(17,21,23,22)(18,19,25,26)(20,29,30,31)(24,27,28,32)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,f^b=fh,e^c=eh,g^c=gh98 conjugacy classes — view the full character table.
| maximal subgroup of | #72306492, #72306502, #72306506, … |
| maximal normal subgroup of | #72306492, #72306502, #72306506, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 101075);