Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232128 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
983 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,12,18,28)(8,21,23,30)(9,16)(10,15)(11,14)(13,24,31,25)(17,29)(19,27,20,32)(22,26)(33,36)(34,35);(1,2)(3,4)(5,7)(6,12)(8,22)(9,18)(10,20)(11,19)(13,29)(14,27)(15,32)(16,28)(17,25)(21,26)(23,34)(24,33)(30,35)(31,36);(2,4)(5,8,11,25,9,23,10,24)(6,13,15,21,16,31,14,30)(7,17,20,22,18,33,19,34)(12,26,27,29,28,35,32,36);(6,14,16,15)(7,18)(12,27,28,32)(13,30,31,21)(17,33)(19,20)(22,34)(26,29,35,36)Polycyclic presentation:
2^9|a^2=h,c^2=ef,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^c=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 291627);