Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×D4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| socle | C22 | C2×B(4,𝔽2) |
| prime radical | C2×#3529 | C2 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
2473 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)(7,15,16,28)(8,12)(9,21,23,13)(10,14)(11,18)(17,29,30,36)(19,32,34,26)(20,27)(22,24)(25,31)(33,35);(5,7)(6,11)(8,17)(9,22)(10,16)(12,25)(13,28)(14,24)(15,21)(18,23)(19,33)(20,30)(26,36)(27,35)(29,32)(31,34);(2,4)(7,16)(11,24)(13,21)(17,30)(18,22)(25,35)(26,32)(31,33);(5,8,10,20)(6,12,14,27)(7,17,16,30)(9,19,23,34)(11,25,24,35)(13,26,21,32)(15,29,28,36)(18,31,22,33)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261075, #72261267, #72261282, … |
| maximal normal subgroup of | #72261075, #72261267, #72261282, … |
| maximal quotient of | #72261903, #72281629, #72282633, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400777);