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 | C2×C42 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1240 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)(21,22,24,28,25,29,32,26)(23,27,31,35,30,34,36,33);(5,7,8,17,10,16,9,15)(6,11,18,20,12,19,13,14)(22,26)(24,32)(27,33)(28,29)(31,36)(34,35);(1,2,3,4)(6,12)(11,19)(13,18)(14,20)(21,23)(22,27)(24,31)(25,30)(26,33)(28,35)(29,34)(32,36);(5,8,10,9)(6,13,12,18)(7,15,16,17)(11,20,19,14)(23,30)(27,34)(31,36)(33,35)Polycyclic presentation:
2^9|a^2=g,b^2=eghi,c^2=fh,d^2=h,e^2=hi,g^2=i,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, 112131);