Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2128 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)(9,10)(11,13)(12,14)(15,16)(19,26)(20,28)(22,31)(23,24)(27,32)(29,30);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,27)(20,24)(21,25)(22,30)(23,28)(26,32)(29,31);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,19,28,29,21,30,20,26)(18,22,23,32,25,27,24,31);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,18)(19,22)(20,24)(21,25)(23,28)(26,31)(27,30)(29,32);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,20,21,28)(18,23,25,24)(19,29,30,26)(22,31,27,32)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,d^2=f,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=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257286, #72257418, #72257577, … |
| maximal normal subgroup of | #72257286, #72257418, #72257577, … |
| maximal quotient of | #72265076 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6939184);