Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,8,9)(7,10,11,12)(13,14)(15,17)(16,18)(19,20)(21,22,26,28,25,29,24,30)(23,31,32,27,34,36,33,35);(2,4)(6,9)(10,12)(13,15,16,19)(14,17,18,20)(21,23,25,34)(22,27,29,35)(24,32,26,33)(28,31,30,36);(2,4)(22,28)(23,32)(24,26)(29,30)(31,36)(33,34);(5,7,8,11)(6,10,9,12)(14,18)(17,20)(22,29)(27,35)(28,30)(31,36)Polycyclic presentation:
2^9|a^2=ei,b^2=h,d^2=efi,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 314898);