Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129627 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C23×Q8 | C23 |
| prime radical | #37187 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C23×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,13,11,16,9,15)(7,17,19,35,20,32,26,34)(8,21,24,31,25,14,23,33)(12,27,29,22,30,36,28,18);(1,2)(3,4)(5,7)(6,12)(8,22)(9,19)(10,26)(11,20)(13,29)(14,32)(15,28)(16,30)(17,21)(18,25)(23,36)(24,27)(31,35)(33,34);(2,4)(6,13)(9,10)(12,28)(14,33)(15,16)(17,34)(19,26)(21,31)(23,24)(27,36)(29,30)(32,35)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dh,e^c=ei,g^c=gi,e^d=ei,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72286501, #72286559, #72286626, … |
| maximal normal subgroup of | #72286501, #72286559, #72286626, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55031);