Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455159 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16260 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1507 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,11,8,12,6,23)(3,13,17,25,18,30,16,9)(4,19,21,26,22,27,5,10)(14,24,15,29,20,32,31,28);(1,3,6,16,8,18,7,17)(2,9,11,13,12,25,23,30)(4,20,5,15,22,14,21,31)(10,24,19,29,26,32,27,28);(1,4)(2,10)(3,14)(5,7)(6,21)(8,22)(9,24)(11,27)(12,26)(13,28)(15,16)(17,31)(18,20)(19,23)(25,32)(29,30);(1,5,7,4,8,21,6,22)(2,10,23,27,12,26,11,19)(3,15,17,20,18,31,16,14)(9,24,30,28,25,32,13,29)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=ei,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72215974, #72216141, #72219691, … |
| maximal normal subgroup of | #72215974, #72216141, #72219691, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362356);