Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,26)(19,24,28,32)(20,25,21,23)(27,30,29,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16);(9,11)(10,13)(12,16)(14,15)(18,23)(20,21)(24,30)(25,26)(27,29)(31,32);(2,5)(6,8)(11,16)(13,15)(17,19,22,28)(18,24,26,32)(20,29,21,27)(23,30,25,31);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,22,21)(18,25,26,23)(19,27,28,29)(24,30,32,31)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=fi,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^c=ei,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
| maximal subgroup of | #72257261, #72257494, #72257641, … |
| maximal normal subgroup of | #72257261, #72257494, #72257641, … |
| maximal quotient of | #72264700 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6905758);