Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395174 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2404 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1204 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)(5,6,9,13,10,14,24,15)(7,16,18,26,20,11,19,27)(8,21,23,12,22,30,36,29)(17,31,33,28,32,35,34,25);(1,2,4,3)(5,7)(6,11)(8,17)(9,19)(10,20)(12,28)(13,26)(14,16)(15,27)(18,24)(21,35)(22,32)(23,34)(25,29)(30,31)(33,36);(5,8)(6,12,14,29)(7,17)(9,23)(10,22)(11,25,16,28)(13,30,15,21)(18,33)(19,34)(20,32)(24,36)(26,35,27,31);(1,3,4,2)(6,13,14,15)(8,22)(11,26,16,27)(12,21,29,30)(17,32)(23,36)(25,31,28,35)(33,34)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,c^2=g,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280086 |
| maximal normal subgroup of | #72280086 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311068);