Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1731 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18)(19,23)(20,29)(21,24)(22,26)(25,30)(27,32)(28,31);(1,3)(2,5)(4,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,27)(18,23,26,32)(20,25,31,24)(21,29,30,28);(2,6)(3,7)(10,13)(14,16)(17,20,30,32,22,31,21,23)(18,19,24,28,26,27,25,29);(9,11,12,15)(10,14,13,16)(19,27)(20,31)(23,32)(28,29);(3,7)(5,8)(17,21,22,30)(18,24,26,25)(19,28,27,29)(20,23,31,32)Polycyclic presentation:
2^9|b^2=i,c^2=fgh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308644, #72308844 |
| maximal normal subgroup of | #72308644, #72308844 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6670741);