Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394463 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#400 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1020 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,14,10,15,11,26)(7,16,18,28,20,31,19,12)(8,21,23,35,24,13,25,30)(17,29,32,36,33,34,22,27);(5,7,11,18,10,20,9,19)(6,12,26,16,15,28,14,31)(8,22,25,17,24,32,23,33)(13,29,35,36,21,34,30,27);(5,8,9,23,10,24,11,25)(6,13,14,30,15,21,26,35)(7,17,18,32,20,33,19,22)(12,27,16,29,28,36,31,34);(2,4)Polycyclic presentation:
2^9|a^2=e,b^2=ef,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72252906, #72256881, #72256910 |
| maximal normal subgroup of | #72252906, #72256881, #72256910 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303988);