Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2459 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14838 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
965 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,5,11,6,19)(3,8,12,24,14,23,15,21)(4,16,20,30,17,9,18,25)(13,27,26,31,28,22,29,32);(1,3)(2,8)(4,13)(5,14)(6,12)(7,15)(9,22)(10,21)(11,23)(16,27)(17,28)(18,26)(19,24)(20,29)(25,31)(30,32);(3,12,14,15)(8,21,23,24)(13,26,28,29)(22,31,27,32);(1,4,5,17)(2,9)(3,13,14,28)(6,18,7,20)(8,22)(10,25)(11,16)(12,26,15,29)(19,30)(21,31)(23,27)(24,32)Polycyclic presentation:
2^9|a^2=eg,c^2=f,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280193 |
| maximal normal subgroup of | #72280193 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309477);