Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417902 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2684 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1955 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)(7,17,20,26)(8,21)(9,14)(10,15)(11,16)(12,18,28,25)(13,23)(19,33,34,27)(22,30)(24,29)(31,32,36,35);(1,2)(3,4)(7,18,20,25)(8,22)(12,26,28,17)(13,29)(19,32,34,35)(21,30)(23,24)(27,36,33,31);(2,4)(5,7)(6,12)(8,19)(9,25)(10,18)(11,20)(13,27)(14,26)(15,17)(16,28)(21,31)(22,34)(23,35)(24,32)(29,33)(30,36);(5,8)(6,13)(7,19)(9,23)(10,24)(11,22)(12,27)(14,30)(15,21)(16,29)(17,31)(18,32)(20,34)(25,35)(26,36)(28,33)Polycyclic presentation:
2^9|a^2=e,b^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=dei,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, 325098);