Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2804 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6)(7,17,22,30)(8,23,27,33)(9,14)(10,15)(11,16)(12,20,31,21)(13,26,34,19)(18,35)(24,36)(25,32)(28,29);(5,7)(6,12)(8,24)(9,27)(10,21)(11,22)(13,14)(15,30)(16,31)(17,32)(18,26)(19,28)(20,25)(23,29)(33,35)(34,36);(1,2,4,3)(5,8,25,26,11,27,10,19)(6,13,32,23,16,34,15,33)(7,18,20,9,22,28,21,24)(12,29,17,36,31,35,30,14);(1,3,4,2)(5,9,25,28,11,24,10,18)(6,14,32,29,16,36,15,35)(7,19,20,8,22,26,21,27)(12,23,17,34,31,33,30,13)Polycyclic presentation:
2^9|a^2=h,c^2=efg,d^2=efg,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280002 |
| maximal normal subgroup of | #72280002 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295271);