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 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,9)(10,12)(13,14,16,17)(15,19,20,18)(21,22,26,28)(23,31,33,29)(24,27,32,36)(25,34,35,30);(1,2,3,4)(5,6,8,9)(7,10,11,12)(14,17)(18,19)(21,23,25,24,26,33,35,32)(22,27,30,31,28,36,34,29);(2,4)(7,11)(10,12)(13,15,16,20)(14,18,17,19)(22,28)(23,32)(24,33)(25,35)(27,31)(29,36);(5,7,8,11)(6,10,9,12)(15,20)(18,19)(21,24,25,33,26,32,35,23)(22,29,30,27,28,31,34,36)Polycyclic presentation:
2^9|a^2=f,b^2=efgh,c^2=fi,d^2=efi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,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, 381007);