Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33568 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×B(4,𝔽2) |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1790 |
| prime radical | #62759 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2398 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,17,5,10)(6,11)(7,12)(8,14,23,19)(9,18,25,20)(15,26)(16,22)(21,28)(24,29,27,31)(30,32);(1,3)(2,8)(4,14)(5,19)(6,16)(7,15)(9,22)(10,26)(11,27)(12,23)(13,17)(18,29)(20,31)(21,30)(24,28)(25,32);(1,4)(2,9)(3,14)(5,7)(6,20)(8,22)(10,11)(12,25)(13,24)(15,19)(16,31)(17,28)(18,21)(23,32)(26,27)(29,30);(3,15)(8,23)(13,26)(14,19)(16,30)(22,32)(24,27)(29,31)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,f^e=fi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261708, #72261720, #72261795, … |
| maximal normal subgroup of | #72261708, #72261720, #72261795, … |
| maximal quotient of | #72340419, #72340667 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400801);