Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #462081 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16416 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
881 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)(3,7,13,22)(4,14,17,8)(5,18,19,9)(6,10)(11,23)(12,27)(15,20)(16,25)(21,29)(24,30)(26,32,28,31);(1,3)(2,7,10,22)(4,15)(5,12)(6,13)(8,23,14,20)(9,21,18,27)(11,17)(16,28)(19,29)(24,32,25,31)(26,30);(1,4)(2,8)(3,11,13,15)(5,16)(6,17)(7,20,22,23)(9,25)(10,14)(12,28,29,26)(18,24)(19,30)(21,31,27,32);(1,5)(2,9)(3,12)(4,16)(6,19)(7,21)(8,24)(10,18)(11,26)(13,29)(14,25)(15,28)(17,30)(20,31)(22,27)(23,32)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=ef,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,d^c=de,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237760, #72237765, #72237768, … |
| maximal normal subgroup of | #72237760, #72237765, #72237768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369277);