Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #164187 |
| 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 | #36147 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1253 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)(7,13,15,10)(8,16)(9,12)(11,17)(14,18,20,19)(21,22)(23,26)(24,27)(25,29)(28,32)(30,33)(31,35)(34,36);(1,2)(3,4)(21,23,25,31)(22,26,29,35)(24,30,32,36)(27,33,28,34);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(22,27)(24,32)(26,33)(28,29)(30,36)(34,35);(2,4)(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fh,d^b=de,f^c=fi,g^c=ghi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 129015);