Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,7,4,8)(18,23)(19,27)(20,21)(25,26)(29,30)(31,32);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,26,22,25,20,23)(19,28,27,31,29,24,30,32);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,19,20,30,22,29,21,27)(18,24,23,31,25,28,26,32);(1,4,5,3)(2,8,6,7)(9,11)(10,14)(12,15)(13,16)(18,25)(19,29)(23,26)(27,30);(11,15)(14,16)(18,25)(23,26)(24,28)(31,32)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=f,d^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257297, #72257516, #72257627 |
| maximal normal subgroup of | #72257297, #72257516, #72257627 |
| maximal quotient of | #72265581 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6464611);