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 | |
995 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)(7,19,22,13,23,32,21,33)(8,24,26,34,27,36,31,14)(9,25,28,35,30,15,29,20);(2,4)(5,7)(6,13)(8,25)(9,24)(10,22)(11,21)(12,23)(14,28)(15,27)(16,32)(17,19)(18,33)(20,26)(29,34)(30,36)(31,35);(1,2,3,4)(5,8,11,31,12,27,10,26)(6,14,17,36,18,34,16,24)(7,20,21,25,23,35,22,15)(9,19,28,33,30,32,29,13);(5,9)(6,15)(7,14)(8,19)(10,29)(11,28)(12,30)(13,26)(16,35)(17,20)(18,25)(21,36)(22,24)(23,34)(27,32)(31,33)Polycyclic presentation:
2^9|a^2=h,c^2=efg,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, 293566);