Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,15)(12,14)(13,16)(18,22)(23,25)(24,26)(30,31);(1,3,5,8)(2,6,7,4)(9,11)(10,13)(12,16)(14,15)(17,18,20,22)(19,26,27,24)(21,23,29,25)(28,31,32,30);(10,14)(13,15)(18,23)(21,29)(22,25)(24,30)(26,31)(28,32);(1,4,5,6)(2,3,7,8)(11,16)(13,15)(17,19,20,27)(18,24,22,26)(21,28,29,32)(23,30,25,31);(18,22)(19,27)(23,25)(28,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,d^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319879 |
| maximal normal subgroup of | #72319879 |
| maximal quotient of | #72320003 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7418358);