Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33551 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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×#453 | C22 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,15)(8,12)(9,13)(10,14)(11,18)(16,28)(17,29)(19,25)(20,26)(21,34)(22,27)(23,31)(24,32)(30,36)(33,35);(5,7,10,17)(6,11,14,24)(8,16,20,30)(9,18,22,32)(12,23,26,35)(13,15,27,29)(19,31,21,33)(25,28,34,36);(5,8,10,20)(6,12,14,26)(7,16,17,30)(9,21,22,19)(11,23,24,35)(13,25,27,34)(15,28,29,36)(18,33,32,31);(1,3,4,2)(7,17)(11,24)(15,29)(16,30)(18,32)(23,35)(28,36)(31,33)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=g,b^a=be,c^a=cf,h^a=hi,d^b=di,f^b=fhi,e^c=eh,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72281983, #72317528, #72317552, … |
| maximal normal subgroup of | #72281983, #72317528, #72317552, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402970);