Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395638 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14725 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2185 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)(3,12)(4,17,19,24)(5,20)(6,10)(7,11)(8,15)(9,18,25,21)(13,26)(14,22,28,32)(16,23)(27,29,31,30);(1,3,5,15,7,13,6,16)(2,8,10,12,11,23,20,26)(4,14,18,29,19,28,21,30)(9,22,24,27,25,32,17,31);(3,13)(5,6)(10,20)(12,26)(14,28)(17,24)(18,21)(27,31);(1,4)(2,9)(3,14)(5,18)(6,21)(7,19)(8,22)(10,24)(11,25)(12,27)(13,28)(15,29)(16,30)(17,20)(23,32)(26,31)Polycyclic presentation:
2^9|a^2=f,b^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,e^c=ei,h^c=hi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72241302, #72241305, #72286431, … |
| maximal normal subgroup of | #72241302, #72241305, #72286431, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302834);