Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232169 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,14,10,15,26,13)(7,16,20,17,21,28,19,11)(8,12,23,31,24,32,25,30)(18,33,35,27,36,29,22,34);(1,2)(3,4)(5,7)(6,11)(8,22)(9,20)(10,21)(12,29)(13,28)(14,16)(15,17)(18,23)(19,26)(24,35)(25,36)(27,30)(31,34)(32,33);(2,4)(5,6)(7,17,21,11)(8,12)(9,13)(10,15)(14,26)(16,19,28,20)(18,34,36,27)(22,33,35,29)(23,30)(24,32)(25,31);(5,8,10,24)(6,12,15,32)(7,18,21,36)(9,25,26,23)(11,27,17,34)(13,31,14,30)(16,33,28,29)(19,35,20,22)Polycyclic presentation:
2^9|a^2=h,c^2=fhi,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296231);