Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388466 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14520 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1179 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,6,12,5,21)(3,13,16,10,17,26,15,8)(4,9,20,24,19,25,18,27)(14,28,31,22,30,32,29,23);(1,3,5,15,6,17,7,16)(2,8,11,13,12,10,21,26)(4,14,18,29,19,30,20,31)(9,23,24,28,25,22,27,32);(1,4)(2,9,12,25)(3,14)(5,20)(6,19)(7,18)(8,22,10,23)(11,27,21,24)(13,28,26,32)(15,31)(16,29)(17,30);(1,3)(2,10)(4,14)(5,16)(6,17)(7,15)(8,12)(9,23)(11,13)(18,31)(19,30)(20,29)(21,26)(22,25)(24,32)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=eh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72338508, #72338518, #72338524, … |
| maximal normal subgroup of | #72338508, #72338518, #72338524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295664);