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 | #2487 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,15,11,16,18,25)(7,17,21,32,22,34,29,12)(8,23,26,28,14,13,33,9)(19,35,36,20,24,27,31,30);(5,7,18,21,11,22,10,29)(6,12,25,17,16,32,15,34)(8,24,33,31,14,19,26,36)(9,27,13,30,28,35,23,20);(5,8,11,14)(6,13,16,23)(7,19)(9,25,28,15)(10,26,18,33)(12,30)(17,35)(20,32)(21,36)(22,24)(27,34)(29,31);(2,4)(5,9,10,23,11,28,18,13)(6,14,15,33,16,8,25,26)(7,20,21,27,22,30,29,35)(12,31,17,19,32,36,34,24)Polycyclic presentation:
2^9|a^2=e,b^2=ef,c^2=f,d^2=e,e^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^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72322821 |
| maximal normal subgroup of | #72322821 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 333475);