Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,16)(15,19,20,18)(21,22,27,32)(23,33,35,29)(24,36,34,28)(25,31,26,30);(13,15,17,20)(14,18,16,19)(21,23,25,34,27,35,26,24)(22,28,30,33,32,36,31,29);(6,9)(10,12)(14,16)(18,19)(21,23,27,35)(22,28,32,36)(24,26,34,25)(29,31,33,30);(2,4)(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,19,20,18)(21,24,25,23,27,34,26,35)(22,29,30,28,32,33,31,36)Polycyclic presentation:
2^9|a^2=egi,b^2=e,c^2=i,d^2=f,e^2=i,f^2=i,g^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,d^b=di,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369569);