Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×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×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1962 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)(7,19,22,32)(8,23,26,35)(9,27)(10,16)(11,17)(12,18)(13,20,33,21)(14,25,31,30)(15,29)(24,34)(28,36);(5,7)(6,13)(8,24)(9,25)(10,21)(11,20)(12,22)(14,34)(15,35)(16,32)(17,19)(18,33)(23,27)(26,28)(29,30)(31,36);(2,4)(5,8)(6,14)(7,9)(10,25)(11,30)(12,26)(13,27)(15,33)(16,23)(17,35)(18,31)(19,34)(20,24)(21,28)(22,29)(32,36);(2,4)(5,9)(6,15)(7,8)(10,28)(11,24)(12,29)(13,31)(14,33)(16,34)(17,36)(18,27)(19,23)(20,30)(21,25)(22,26)(32,35)Polycyclic presentation:
2^9|a^2=h,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.
| maximal subgroup of | #72258208, #72258209, #72258553, … |
| maximal normal subgroup of | #72258208, #72258209, #72258553, … |
| maximal quotient of | #72239302, #72258236, #72258245, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290259);