Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #386374 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14427 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1059 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,8,5,11)(3,13,14,22,16,24,21,9)(4,17,18,10,20,26,19,23)(15,25,28,27,30,31,29,32);(2,8)(3,14)(5,6)(9,24)(13,22)(15,29)(16,21)(17,26)(18,19)(25,32)(27,31)(28,30);(1,3)(2,9,8,22)(4,15)(5,14)(6,21)(7,16)(10,25,23,31)(11,13,12,24)(17,27,26,32)(18,29)(19,28)(20,30);(1,4)(2,10)(3,15)(5,19)(6,18)(7,20)(8,23)(9,25)(11,17)(12,26)(13,27)(14,28)(16,30)(21,29)(22,31)(24,32)Polycyclic presentation:
2^9|a^2=h,c^2=efg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237047, #72237071, #72237073, … |
| maximal normal subgroup of | #72237047, #72237071, #72237073, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293572);