Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #451053 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15984 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,13,8,14,7,12)(3,15,17,11,18,25,5,9)(4,19,23,10,22,28,21,29)(16,24,20,32,30,27,31,26);(1,3)(2,9)(4,20)(5,7)(6,17)(8,18)(10,26)(11,14)(12,25)(13,15)(16,21)(19,27)(22,31)(23,30)(24,28)(29,32);(1,4)(2,10)(3,16)(5,20)(6,21)(7,23)(8,22)(9,24)(11,27)(12,28)(13,19)(14,29)(15,26)(17,31)(18,30)(25,32);(1,5,7,3,8,17,6,18)(2,11,13,15,14,9,12,25)(4,20,23,16,22,31,21,30)(10,27,19,26,29,24,28,32)Polycyclic presentation:
2^9|a^2=gh,d^2=ei,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237067, #72237072, #72237086, … |
| maximal normal subgroup of | #72237067, #72237072, #72237086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358250);