Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2987 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,17,18)(15,19,23,21)(16,20,24,25)(22,26,28,27);(5,7,9,8)(6,10,11,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27);(2,4)(14,18)(15,21)(16,20)(19,23)(22,28)(24,25);(5,8,9,7)(6,10,11,12)(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28);(13,14)(17,18)(19,21)(20,25)(22,26)(27,28)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72256962, #72256969, #72257012, … |
| maximal normal subgroup of | #72256962, #72256969, #72257012, … |
| maximal quotient of | #72265055, #72265143 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483728);