Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
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 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,9)(7,12,8,11)(13,14,16,18)(15,17,19,20)(21,22,27,32)(23,33,35,30)(24,36,34,28)(25,31,26,29);(5,7,10,8)(6,11,9,12)(13,15,16,19)(14,17,18,20)(21,23,26,24,27,35,25,34)(22,28,31,30,32,36,29,33);(1,2,3,4)(5,8,10,7)(6,11,9,12)(15,19)(17,20)(22,29)(23,24)(25,26)(30,33)(31,32)(34,35);(5,9,10,6)(7,12,8,11)(21,24,26,35,27,34,25,23)(22,30,31,36,32,33,29,28)Polycyclic presentation:
2^9|a^2=eg,b^2=eghi,c^2=ef,d^2=ei,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, 381371);