Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454975 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16258 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1496 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)(3,12,15,7)(4,8)(5,19)(6,10)(9,18)(11,22)(13,23,31,30)(14,24,32,26)(16,25,20,29)(17,28)(21,27);(1,3)(2,7)(4,14)(5,20)(6,15)(8,26)(9,24)(10,12)(11,29)(13,17)(16,22)(18,32)(19,25)(21,31)(23,27)(28,30);(1,4)(2,8)(3,13)(5,21)(6,18)(7,23)(9,10)(11,28)(12,30)(14,20)(15,31)(16,32)(17,22)(19,27)(24,25)(26,29);(1,4)(2,9,10,8)(3,14)(5,17)(6,18)(7,24,12,26)(11,28,19,27)(13,20)(15,32)(16,31)(21,22)(23,29,30,25)Polycyclic presentation:
2^9|a^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,d^c=de,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72240432, #72240445, #72240448, … |
| maximal normal subgroup of | #72240432, #72240445, #72240448, … |
| maximal quotient of | #72251292, C2×#454975, #72255663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362171);