Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418769 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
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 | |
1923 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)(8,23,26,33)(9,16)(10,15)(11,17)(12,20)(13,21)(14,27,32,25)(19,30,35,28)(22,29)(24,31,36,34);(5,7)(6,12)(8,24)(9,21)(10,20)(11,22)(13,17)(14,31)(15,29)(16,18)(19,25)(23,28)(26,36)(27,35)(30,33)(32,34);(2,4)(5,7)(6,13)(8,19)(9,21)(10,20)(11,22)(12,17)(14,30)(15,18)(16,29)(23,31)(24,27)(25,36)(26,35)(28,32)(33,34);(5,8)(6,14)(7,19)(9,27)(10,25)(11,26)(12,28)(13,30)(15,33)(16,23)(17,32)(18,34)(20,36)(21,24)(22,35)(29,31)Polycyclic presentation:
2^9|a^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326048);