Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996335 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2664 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)(7,11)(10,12)(13,14,18,19,17,22,15,20)(16,24,26,28,25,21,23,27);(6,9)(7,11)(14,19)(15,18)(16,23)(20,22)(25,26)(27,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,18)(14,20,22,19)(16,23,25,26)(21,28,24,27);(6,9)(10,12)(13,16,18,26,17,25,15,23)(14,21,19,27,22,24,20,28);(5,7,8,11)(6,10,9,12)(13,15,17,18)(14,19,22,20)(16,23,25,26)(21,27,24,28)Polycyclic presentation:
2^9|a^2=fi,c^2=i,d^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72280988, #72284485, #72285031, … |
| maximal normal subgroup of | #72280988, #72284485, #72285031, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7052936);