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 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2187 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)(7,11)(8,10)(9,12)(14,19)(15,23)(16,24)(22,27);(5,7,9,8)(6,10,12,11)(13,14,17,19)(15,21,23,28)(16,25,24,20)(18,27,26,22);(2,4)(5,7,9,8)(6,11,12,10)(13,15,17,23)(14,20,19,25)(16,26,24,18)(21,22,28,27);(5,8,9,7)(6,10,12,11)(15,16)(18,26)(20,28)(21,25)(22,27)(23,24);(13,16,17,24)(14,21,19,28)(15,18,23,26)(20,27,25,22)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=fi,e^2=i,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 | #72257154, #72257166, #72257217, … |
| maximal normal subgroup of | #72257154, #72257166, #72257217, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484361);