Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31418 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14417 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1371 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,3,12,10)(4,8,16,25,17,13,21,24)(5,18,19,14,20,26,22,9)(15,28,30,23,31,29,32,27);(1,3)(2,7)(4,13)(5,18)(6,10)(8,17)(9,22)(11,12)(14,19)(15,29)(16,24)(20,26)(21,25)(23,32)(27,30)(28,31);(1,4,12,16,7,17,6,21)(2,8,11,24,3,13,10,25)(5,15,19,32,20,31,22,30)(9,23,26,29,14,27,18,28);(1,5)(2,9)(3,14)(4,15)(6,22)(7,20)(8,23)(10,18)(11,26)(12,19)(13,27)(16,32)(17,31)(21,30)(24,29)(25,28)Polycyclic presentation:
2^9|a^2=h,c^2=eg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319366, #72319370, #72319460 |
| maximal normal subgroup of | #72319366, #72319370, #72319460 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293130);