Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201085 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2433 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,5,7,8)(4,6)(9,10,14,16,13,11,12,15);(1,3,4,7)(2,5)(6,8)(9,11,13,10)(12,15,14,16);(12,14)(15,16);(3,7)(5,8)Polycyclic presentation:
2^9|a^2=g,b^2=egi,e^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=ghi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72231446, #72231451, #72231454, … |
| maximal normal subgroup of | #72231446, #72231451, #72231454, … |
| maximal quotient of | #72232716, #72232766, #72232773, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108489);