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 | |
1036 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)(15,19)(16,17)(18,20)(21,22,27,33)(23,30,24,31)(25,36,35,29)(26,34,28,32);(1,2,3,4)(5,6,8,9)(7,10,11,12)(14,17)(18,19)(21,23,26,35,27,24,28,25)(22,29,32,30,33,36,34,31);(2,4)(7,11)(10,12)(13,15)(14,18)(16,20)(17,19)(21,24,27,23)(22,30,33,31)(25,26,35,28)(29,34,36,32);(5,7,8,11)(6,10,9,12)(15,20)(18,19)(21,25,26,23,27,35,28,24)(22,31,32,29,33,30,34,36)Polycyclic presentation:
2^9|a^2=i,b^2=efgh,c^2=i,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, 380554);