Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #161570 |
| 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 | #2856 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1140 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)(8,12)(9,18)(10,13)(11,16)(15,19)(17,20)(21,22)(23,26,30,33)(24,27)(25,28)(29,32,35,36)(31,34);(5,7,9,16,10,17,8,15)(6,11,12,14,13,19,18,20)(21,23)(22,26,28,33)(24,29)(25,30)(27,32,34,36)(31,35);(1,2,3,4)(24,31)(27,34)(29,35)(32,36);(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,14,19,20)(21,24,25,31)(22,27,28,34)(23,29,30,35)(26,32,33,36)Polycyclic presentation:
2^9|a^2=g,b^2=eg,c^2=f,d^2=h,e^2=hi,b^a=be,c^a=cf,e^a=eh,e^b=ei,g^b=gi,d^c=di98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111823);