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 | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
993 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)(5,6,11,26,10,16,12,17)(7,18,20,29,9,13,21,15)(8,14,27,31,24,33,25,32)(19,34,35,36,23,28,22,30);(5,7,10,9)(6,13,16,18)(8,22,24,35)(11,21,12,20)(14,30,33,36)(15,26,29,17)(19,27,23,25)(28,32,34,31);(5,8,10,24)(6,14)(7,19,9,23)(11,27,12,25)(13,28)(15,30)(16,33)(17,32)(18,34)(20,35,21,22)(26,31)(29,36);(2,4)(5,9,10,7)(6,15,16,29)(8,23,24,19)(11,20,12,21)(13,17,18,26)(14,30,33,36)(22,27,35,25)(28,32,34,31)Polycyclic presentation:
2^9|a^2=eg,b^2=i,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, 341165);