Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1147 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(19,27,30,29)(23,31,32,28);(2,6)(7,8)(10,13)(14,15)(17,18)(19,28,30,31)(20,25)(21,26)(22,24)(23,29,32,27);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,19)(18,23)(20,29)(21,27)(22,30)(24,32)(25,28)(26,31);(1,4,5,3)(2,7,6,8)(10,13)(14,15)(18,24)(23,32)(25,26)(28,31)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=e,d^2=efg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279999);