Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418473 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2673 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15369 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1299 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,7,11,6,21)(3,12,14,23,15,25,20,8)(4,16,18,9,19,27,22,28)(13,29,17,32,30,24,31,26);(1,3)(2,8)(4,17)(5,20)(6,14)(7,15)(9,26)(10,25)(11,23)(12,21)(13,18)(16,29)(19,31)(22,30)(24,27)(28,32);(8,23)(12,25)(24,29)(26,32);(1,4)(2,9)(3,13,15,30)(5,18)(6,22)(7,19)(8,24,23,29)(10,27)(11,28)(12,26,25,32)(14,17,20,31)(16,21)Polycyclic presentation:
2^9|a^2=e,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72237063, #72237070, #72237086, … |
| maximal normal subgroup of | #72237063, #72237070, #72237086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334591);