Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #391213 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14621 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1118 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,8,5,18)(3,12,14,9,15,24,19,21)(4,10,16,27,17,22,20,25)(13,28,31,23,30,32,29,26);(2,8)(4,16)(5,6)(10,25)(12,24)(13,29)(14,19)(17,20)(22,27)(23,28)(26,32)(30,31);(1,3)(2,9)(4,13)(5,19)(6,14)(7,15)(8,21)(10,26)(11,24)(12,18)(16,31)(17,30)(20,29)(22,23)(25,32)(27,28);(1,4)(2,10)(3,13)(5,20)(6,16)(7,17)(8,22)(9,23)(11,27)(12,28)(14,31)(15,30)(18,25)(19,29)(21,26)(24,32)Polycyclic presentation:
2^9|a^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,f^c=fi,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237047, #72237067, #72237069, … |
| maximal normal subgroup of | #72237047, #72237067, #72237069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298411);