Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | H(3,𝔽4) |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2782 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,17,12,18,10,16)(7,19,23,13,24,26,22,33)(8,25,27,14,29,34,28,21)(9,15,30,20,32,36,31,35);(1,2)(3,4)(5,7)(6,13)(8,15)(9,25)(10,23)(11,22)(12,24)(14,31)(16,26)(17,19)(18,33)(20,28)(21,30)(27,35)(29,36)(32,34);(2,4)(5,8,10,27,12,29,11,28)(6,14,16,34,18,21,17,25)(7,20,23,15,24,35,22,36)(9,26,31,33,32,19,30,13);(5,9)(6,15)(7,21)(8,26)(10,31)(11,30)(12,32)(13,28)(14,24)(16,35)(17,20)(18,36)(19,29)(22,34)(23,25)(27,33)Polycyclic presentation:
2^9|a^2=h,c^2=efh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296354);