Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #162342 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,14)(8,13)(9,18)(10,12)(11,15)(16,20)(17,19)(21,22)(23,26,30,33)(24,28)(25,27)(29,34,31,36)(32,35);(5,7,9,15,10,16,8,17)(6,11,13,14,12,19,18,20)(21,23)(22,26,27,33)(24,31)(25,30)(28,36,35,34)(29,32);(1,2,3,4)(6,12)(11,19)(13,18)(14,20)(22,27)(24,32)(26,33)(29,31);(5,8,10,9)(6,13,12,18)(7,15,16,17)(11,20,19,14)(21,24,25,32)(22,28,27,35)(23,29,30,31)(26,34,33,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=fh,d^2=h,e^2=hi,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,g^b=gi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111840);