Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 | #24438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(10,14)(11,12)(17,18,21,24)(19,27,30,23)(20,26,22,25)(28,31,29,32);(9,10,13,14)(11,15,12,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,27)(25,32)(26,31);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(18,24)(19,28)(20,22)(23,31)(27,32)(29,30);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,20,21,22)(18,25,24,26)(19,29,30,28)(23,32,27,31);(10,14)(15,16)(17,20)(18,26)(19,30)(21,22)(23,27)(24,25)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fi,d^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72251197, #72256835, #72256909, … |
| maximal normal subgroup of | #72251197, #72256835, #72256909, … |
| maximal quotient of | #72265554 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6605404);