Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994043 |
| 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 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1748 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)(11,13)(12,14)(15,16)(17,18)(19,27)(20,24)(21,31)(22,26)(23,29)(25,28)(30,32);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,28,30,22,20,21,29)(18,23,31,24,26,32,25,27);(2,5)(6,8)(9,11,12,16)(10,13,14,15);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(17,20,22,19)(18,24,26,27)(21,29,28,30)(23,25,32,31);(10,14)(13,15)(19,20)(23,32)(24,27)(29,30)Polycyclic presentation:
2^9|b^2=fgi,c^2=h,d^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei,f^d=fi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72285261 |
| maximal normal subgroup of | #72285261 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6988773);