Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201085 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#2541 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,6,8,5,3,4,7)(9,10,13,16,14,15,12,11);(1,3,5,2)(4,7,6,8)(9,11,13,15,14,16,12,10);(4,6)(7,8);(9,12,14,13)(10,11,15,16)Polycyclic presentation:
2^9|a^2=g,b^2=eghi,d^2=h,e^2=hi,g^2=i,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 | #72214595, #72214707, #72214750, … |
| maximal normal subgroup of | #72214595, #72214707, #72214750, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108401);