Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418818 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15506 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1043 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,13,8,11)(3,14,18,28,19,29,17,9)(4,5,22,24,21,25,10,26)(15,31,20,32,30,23,27,16);(1,3)(2,9)(4,20)(5,23)(6,18)(7,19)(8,17)(10,15)(11,29)(12,14)(13,28)(16,24)(21,27)(22,30)(25,31)(26,32);(1,4,8,22,7,21,6,10)(2,5,11,24,13,25,12,26)(3,15,17,20,19,30,18,27)(9,16,29,31,28,32,14,23);(1,5)(2,10)(3,16)(4,11)(6,26)(7,25)(8,24)(9,27)(12,21)(13,22)(14,30)(15,29)(17,31)(18,23)(19,32)(20,28)Polycyclic presentation:
2^9|a^2=fg,c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319370, #72319383, #72319460, … |
| maximal normal subgroup of | #72319370, #72319383, #72319460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341921);