Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33551 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#3540 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2461 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)(7,17,20,32)(8,13)(9,14)(10,24,26,15)(11,16)(12,19)(18,34,22,33)(21,31,23,29)(25,28)(27,35)(30,36);(5,7)(6,12)(8,18)(9,22)(10,25)(11,20)(13,27)(14,30)(15,32)(16,28)(17,24)(19,26)(21,36)(23,35)(29,33)(31,34);(2,4)(5,8,11,9)(6,13,16,14)(7,18,20,22)(10,21,26,23)(12,27,28,30)(15,31,24,29)(17,33,32,34)(19,35,25,36);(5,9,11,8)(6,14,16,13)(7,18,20,22)(10,23,26,21)(12,27,28,30)(15,31,24,29)(17,34,32,33)(19,35,25,36)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261251, #72261282, #72261294, … |
| maximal normal subgroup of | #72261251, #72261282, #72261294, … |
| maximal quotient of | #72261831 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400782);