Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #184979 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×H(5,𝔽2) | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
17290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,14,16,11)(8,10)(9,13)(12,17)(15,19,20,18);(6,10)(8,17)(11,18)(12,13)(14,19)(15,20);(5,7)(6,11)(8,15)(9,16)(10,18)(12,19)(13,14)(17,20);(2,4)Polycyclic presentation:
2^9|a^2=h,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,e^b=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72210545, #72213597, #72213729, … |
| maximal normal subgroup of | #72210545, #72213597, #72213729, … |
| maximal quotient of | #72213729, #72213738, C2×#392132, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299328);