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 | #2487 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1258 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)(5,6,11,15,9,13,10,23)(7,16,18,27,19,12,24,30)(8,20,25,28,21,35,22,14)(17,31,33,29,32,34,36,26);(2,4)(5,7)(6,12)(8,17)(9,19)(10,18)(11,24)(13,16)(14,26)(15,27)(20,34)(21,32)(22,33)(23,30)(25,36)(28,29)(31,35);(6,13)(8,21)(15,23)(17,32)(22,25)(26,29)(31,34)(33,36);(5,8,10,22,9,21,11,25)(6,14,15,20,13,28,23,35)(7,17,18,33,19,32,24,36)(12,26,27,34,16,29,30,31)Polycyclic presentation:
2^9|a^2=efh,d^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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 | #72281319, #72281343, #72281398, … |
| maximal normal subgroup of | #72281319, #72281343, #72281398, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 322161);