Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×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 | |
1798 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)(7,18)(8,23,26,14)(9,16)(10,15)(11,17)(12,20)(13,21)(19,31,34,36)(22,29)(24,30,35,28)(25,32,27,33);(5,7)(6,12)(8,19)(9,21)(10,20)(11,22)(13,17)(14,28)(15,29)(16,18)(23,30)(24,27)(25,35)(26,34)(31,32)(33,36);(2,4)(5,7)(6,13)(8,24)(9,21)(10,20)(11,22)(12,17)(14,31)(15,18)(16,29)(19,25)(23,36)(26,35)(27,34)(28,33)(30,32);(5,8)(6,14)(7,19)(9,27)(10,25)(11,26)(12,28)(13,30)(15,33)(16,32)(17,23)(18,31)(20,35)(21,24)(22,34)(29,36)Polycyclic presentation:
2^9|a^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361876);