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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,8,9)(7,10,11,12)(13,14)(15,18)(16,19)(17,20)(21,22)(23,33)(24,36)(25,29)(26,31)(27,32)(28,34)(30,35);(6,9)(10,12)(13,15,17,16)(14,18,20,19)(21,23,25,34,27,35,26,24)(22,28,31,33,32,36,29,30);(1,2,4,3)(6,9)(7,11)(13,15,17,16)(14,19,20,18)(22,29)(23,24)(25,26)(28,36)(31,32)(34,35);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,16,17,15)(14,19,20,18)(21,24,25,23,27,34,26,35)(22,30,31,28,32,33,29,36)Polycyclic presentation:
2^9|a^2=egi,b^2=efh,c^2=efi,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, 378516);