Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,8,10)(7,9,11,12)(14,17)(18,20)(21,22,26,30,25,29,24,28)(23,31,34,27,32,36,33,35);(5,7,8,11)(6,9,10,12)(13,14)(15,18)(16,17)(19,20)(23,32)(27,35)(31,36)(33,34);(2,4)(7,11)(9,12)(14,17)(15,19)(21,23,25,32)(22,27,29,35)(24,34,26,33)(28,36,30,31);(13,15,16,19)(14,18,17,20)(22,28)(24,26)(27,36)(29,30)(31,35)(33,34)Polycyclic presentation:
2^9|a^2=efg,b^2=egi,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374319);