Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2580 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
785 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)(5,6,11,16)(7,18)(8,24,26,13)(9,14,29,27)(10,15,19,17)(12,30)(20,34)(21,35)(22,33)(23,32)(25,28)(31,36);(5,7,19,22,11,23,10,30)(6,12,17,18,16,33,15,32)(8,25,27,20,26,31,14,35)(9,28,13,34,29,36,24,21);(5,8,19,27,11,26,10,14)(6,13,17,29,16,24,15,9)(7,20,22,31,23,35,30,25)(12,28,18,34,33,36,32,21);(2,4)(5,9,10,24,11,29,19,13)(6,14,15,26,16,27,17,8)(7,21,22,28,23,34,30,36)(12,31,18,35,33,25,32,20)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=ef,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322900 |
| maximal normal subgroup of | #72322900 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 334361);