Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453966 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1140 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,9)(7,12,8,11)(13,14)(15,17)(16,18)(19,20)(23,29)(24,30)(26,32)(28,33)(31,35)(34,36);(1,2,3,4)(5,7,10,8)(6,11,9,12)(13,15,16,19)(14,17,18,20)(21,22)(23,26)(24,28)(25,27)(29,32)(30,33)(31,34)(35,36);(2,4)(5,8,10,7)(6,11,9,12)(21,23,30,35,25,31,24,29)(22,26,33,36,27,34,28,32);(5,9,10,6)(7,12,8,11)(14,18)(17,20)(22,27)(26,34)(28,33)(32,36)Polycyclic presentation:
2^9|a^2=e,b^2=egh,c^2=efi,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 379708);