Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2333 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(11,15)(14,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,31)(29,32);(1,3)(2,6)(4,7)(5,8)(10,13)(11,15);(1,2,5,4)(3,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,28,22,29,20,27)(18,23,24,26,25,31,30,32);(1,4,5,2)(3,6,8,7)(10,13)(14,16)(17,18)(19,23)(20,30)(21,24)(22,25)(26,28)(27,32)(29,31);(9,11,12,15)(10,14,13,16)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,26,31,32)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72251190, #72251211, #72256978, … |
| maximal normal subgroup of | #72251190, #72251211, #72256978, … |
| maximal quotient of | #72265279, C2×#7156947 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7064143);