Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460129 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16357 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23407 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1195 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)(3,7)(4,9)(5,18,21,10)(6,14)(8,11)(12,23)(13,20,28,22)(15,30,31,24)(16,25)(17,26)(19,27,32,29);(1,3,7,2,8,14,6,11)(4,9,23,25,17,26,16,12)(5,13,20,10,21,28,22,18)(15,24,19,27,31,30,32,29);(1,4)(2,9)(3,12)(5,19,21,32)(6,16)(7,23)(8,17)(10,27,18,29)(11,26)(13,24,28,30)(14,25)(15,20,31,22);(1,5,6,22,8,21,7,20)(2,10,3,13,11,18,14,28)(4,15,16,32,17,31,23,19)(9,24,12,29,26,30,25,27)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=f,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,g^c=gi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237064, #72237067, #72237069, … |
| maximal normal subgroup of | #72237064, #72237067, #72237069, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367326);