Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2476 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15958 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,7,12,6,10)(3,13,16,8,17,26,22,14)(4,9,20,18,21,25,19,24)(15,27,32,28,31,29,30,23);(1,2)(3,14,17,8)(4,18)(5,11)(6,10)(7,12)(9,19)(13,22,26,16)(15,29,31,27)(20,25)(21,24)(23,30,28,32);(1,3)(2,8)(4,15)(5,16)(6,22)(7,17)(9,23)(10,13)(11,26)(12,14)(18,27)(19,30)(20,32)(21,31)(24,29)(25,28);(1,4,5,19,7,21,6,20)(2,9,10,18,12,25,11,24)(3,15,16,30,17,31,22,32)(8,23,13,27,14,28,26,29)Polycyclic presentation:
2^9|a^2=eh,b^2=ef,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,f^c=fi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319162, #72319383, #72319399, … |
| maximal normal subgroup of | #72319162, #72319383, #72319399, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 357333);