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 | #2679 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,17,21,30)(8,22,18,33)(9,14)(10,15)(11,16)(12,19,31,20)(13,25,34,24)(23,35)(26,32)(27,29)(28,36);(5,7)(6,12)(8,23)(9,25)(10,20)(11,21)(13,32)(14,33)(15,30)(16,31)(17,36)(18,26)(19,28)(22,29)(24,27)(34,35);(1,3,4,2)(5,8,11,18)(6,13,16,34)(7,9,21,27)(10,24,28,25)(12,29,31,14)(15,33,36,22)(17,35,30,32)(19,26,20,23);(5,9)(6,14)(7,18)(8,21)(10,23)(11,27)(12,13)(15,35)(16,29)(17,22)(19,24)(20,25)(26,28)(30,33)(31,34)(32,36)Polycyclic presentation:
2^9|a^2=h,c^2=efh,e^2=i,f^2=i,g^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, 296352);