Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #163012 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1104 conjugacy classes of subgroups; only the 214 normal subgroups are shown.
This group has 1104 conjugacy classes of subgroups — too many to draw. Only its 214 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)(3,7)(4,6)(5,8)(9,10,12,16,13,15,21,23)(11,17,19,24,18,14,20,22)(25,26)(27,34,36,39)(28,37)(29,32)(30,33)(31,35,40,38);(1,3)(2,5)(4,8)(6,7)(11,18)(14,17)(19,20)(22,24);(3,8)(5,7)(9,11,13,18)(10,14,15,17)(12,20,21,19)(16,24,23,22)(25,27)(26,31)(28,35)(29,38)(30,36)(32,39)(33,40)(34,37);(2,6)(5,7)(25,28,30,29)(26,32,33,37)(27,35,36,38)(31,39,40,34)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ceh,f^c=fi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72238884, #72238889 |
| maximal normal subgroup of | #72238884, #72238889 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120824);