Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395174 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2580 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
992 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)(5,6,9,12,10,14,24,15)(7,16,18,27,20,11,19,25)(8,21,23,13,22,28,36,29)(17,31,33,26,32,35,34,30);(1,2,4,3)(5,7,9,19,10,20,24,18)(6,11,12,27,14,16,15,25)(8,17,23,34,22,32,36,33)(13,26,28,31,29,30,21,35);(1,3,4,2)(6,12,14,15)(8,22)(11,25,16,27)(13,21,29,28)(17,32)(23,36)(26,31,30,35)(33,34);(5,8,9,23,10,22,24,36)(6,13,15,21,14,29,12,28)(7,17,19,34,20,32,18,33)(11,26,25,35,16,30,27,31)Polycyclic presentation:
2^9|a^2=ef,b^2=eh,c^2=e,d^2=h,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 | #72249923, #72249941, #72249969, … |
| maximal normal subgroup of | #72249923, #72249941, #72249969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323566);