Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #301949 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62270 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1182 conjugacy classes of subgroups; only the 152 normal subgroups are shown.
This group has 1182 conjugacy classes of subgroups — too many to draw. Only its 152 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,26)(13,19)(14,20)(15,23)(17,27)(18,21)(22,36)(24,30)(25,32)(28,33)(29,39)(31,37)(34,35)(38,40);(1,3)(2,5)(4,8)(6,7)(9,11,21,24,14,25,13,23)(10,15,19,32,20,30,18,16)(12,22,35,38,28,39,27,37)(17,29,33,40,34,36,26,31);(1,2)(3,7,8,5)(4,6)(9,11,14,25)(10,16,20,32)(12,22,28,39)(13,23,21,24)(15,18,30,19)(17,31,34,40)(26,36,33,29)(27,37,35,38);(9,12,21,35,14,28,13,27)(10,17,19,33,20,34,18,26)(11,22,24,38,25,39,23,37)(15,29,32,40,30,36,16,31)Polycyclic presentation:
2^9|b^2=eh,c^2=fh,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,g^c=ghi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 209152);