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 | |
1074 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)(15,17)(16,18)(19,20)(21,22,25,28)(23,30,24,27)(26,29,32,35)(31,36,33,34);(5,7,10,8)(6,11,9,12)(13,15,16,19)(14,17,18,20)(21,23,26,31,25,24,32,33)(22,27,29,34,28,30,35,36);(1,2,3,4)(5,8,10,7)(6,11,9,12)(21,24)(22,27)(23,25)(26,31)(28,30)(29,36)(32,33)(34,35);(5,9,10,6)(7,12,8,11)(14,18)(17,20)(22,28)(23,24)(29,35)(31,33)Polycyclic presentation:
2^9|a^2=e,b^2=egh,c^2=efi,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379705);