Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25443 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1142 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,17)(8,13,24,31)(9,14)(10,15)(11,16)(12,20)(18,33,34,28)(19,32)(21,27)(22,26,35,36)(23,29,25,30);(1,2)(3,4)(5,7)(6,12,16,27)(8,22,24,35)(9,20)(10,21)(11,19)(13,28)(14,32,15,17)(18,23,34,25)(26,30)(29,36)(31,33);(2,4)(5,8,9,23,11,24,10,25)(6,13,15,30,16,31,14,29)(7,18,20,22,19,34,21,35)(12,26,17,33,27,36,32,28);(6,14,16,15)(7,19)(12,17,27,32)(13,29,31,30)(18,34)(20,21)(22,35)(26,33,36,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,d^2=fh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72280601, #72280718 |
| maximal normal subgroup of | #72280601, #72280718 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285238);