Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
974 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22,24,27,25,28,33,36)(23,29,31,34,30,35,32,26);(1,3,4,2)(14,17)(15,19)(21,23)(22,26)(24,32)(25,30)(27,35)(28,34)(29,36)(31,33);(6,9)(10,12)(13,15,16,19)(14,18,17,20)(23,30)(26,34)(29,35)(31,32);(5,7,8,11)(6,10,9,12)(14,17)(18,20)(22,27)(24,33)(26,35)(28,36)(29,34)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=egh,c^2=gi,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,e^b=ei,h^b=hi,e^d=ei,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277901);