Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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 | |
2227 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2)(3,4)(6,9)(7,11);(5,6)(7,10)(8,9)(11,12)(13,14,19,20)(15,16,17,18);(2,4)(13,15,18,14,19,17,16,20);(5,7,8,11)(6,10,9,12)(13,16,19,18)(14,17,20,15);(6,9)(10,12)(13,17,18,20,19,15,16,14)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=fi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256932, #72257039, #72257047, … |
| maximal normal subgroup of | #72256932, #72257039, #72257047, … |
| maximal quotient of | #72264880, #72265089 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715405);