Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458124 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2731 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16635 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
730 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7,6,11,20,10)(3,8,13,23,14,25,19,21)(4,15,17,28,18,9,26,22)(12,24,16,30,29,32,31,27);(2,7,11,10)(8,23,25,21)(9,22,15,28)(24,30,32,27);(1,3)(2,8,11,25)(4,16)(5,19)(6,14)(7,21,10,23)(9,27,15,30)(12,17)(13,20)(18,31)(22,32,28,24)(26,29);(1,4)(2,9,11,15)(3,12,14,29)(5,17)(6,18)(7,22,10,28)(8,24)(13,16,19,31)(20,26)(21,27)(23,30)(25,32)Polycyclic presentation:
2^9|a^2=eg,b^2=g,c^2=g,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72279915 |
| maximal normal subgroup of | #72279915 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 376132);