Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418863 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1047 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,9,24,11,12,10,15)(7,16,25,27,19,13,17,29)(8,20,21,14,23,30,22,26)(18,33,32,31,35,36,34,28);(6,12)(7,17)(9,10)(13,27)(16,29)(18,34)(19,25)(20,30)(21,22)(28,33)(31,36)(32,35);(5,7)(6,13,12,16)(8,18)(9,25)(10,17)(11,19)(14,28,26,31)(15,27,24,29)(20,36,30,33)(21,32)(22,34)(23,35);(2,4)(5,8,11,23)(6,14,12,26)(7,18,19,35)(9,22,10,21)(13,28,16,31)(15,30,24,20)(17,32,25,34)(27,33,29,36)Polycyclic presentation:
2^9|a^2=eg,c^2=ef,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341163);