Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | #24257 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1948 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,24)(25,32)(26,29)(27,30)(28,31);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,20,27)(18,22,23,30)(21,25,28,26)(24,32,31,29);(9,12,13,11)(10,14,15,16)(18,23)(19,25)(21,28)(22,29)(26,27)(30,32);(19,26)(21,28)(22,29)(24,31)(25,27)(30,32);(1,3,4,7)(2,6,5,8)(19,27)(22,30)(25,26)(29,32)Polycyclic presentation:
2^9|b^2=g,c^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72239589, #72251194, #72251237, … |
| maximal normal subgroup of | #72239589, #72251194, #72251237, … |
| maximal quotient of | #72239733, #72264301, #72264925, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354653);