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 | |
858 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)(5,6,8,9)(7,10,11,12)(13,14,16,18)(15,17,19,20)(21,22)(23,30)(24,35)(25,31)(26,29)(27,32)(28,33)(34,36);(6,9)(10,12)(13,15)(14,17)(16,19)(18,20)(21,23,25,33,27,34,26,24)(22,28,29,30,32,35,31,36);(1,2,3,4)(6,9)(7,11)(15,19)(17,20)(22,29)(23,24)(25,26)(30,36)(31,32)(33,34);(5,7,8,11)(6,10,9,12)(21,24,25,23,27,33,26,34)(22,30,29,35,32,36,31,28)Polycyclic presentation:
2^9|a^2=egi,b^2=h,c^2=ef,d^2=e,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, 378071);