Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #92835 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1270 |
| commutator | C4×C8 | C2×C8 |
| Frattini, cosocle | #1744 | C22 |
| prime radical | C23 | #327 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1744 ◁ G | |
| LECS | G ▷ #1744 ▷ C2×C42 ▷ C22 ▷ C1 | |
179 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 179 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2,4,9,6,10,14,16,7,11,15,13,5,3,12,8)(17,18,21,27,23,30,39,26,24,28,38,45,22,29,20,37)(19,31,33,42,35,47,32,43,36,46,48,25,34,41,40,44);(1,3,7,10)(2,5,11,6)(4,8,15,16)(9,12,13,14)(17,19,24,36)(18,25,28,42)(20,33,39,48)(21,40,38,32)(22,35,23,34)(26,41,37,47)(27,31,45,46)(29,44,30,43)Polycyclic presentation:
2^9|a^2=d,b^2=h,c^2=fh,d^2=g,e^2=i,f^2=h,g^2=h,b^a=bc,c^a=ce,e^a=eh,f^a=fi,c^b=cf,d^b=defi,e^b=ei,f^b=fh,g^b=ghi,d^c=dhi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72210225, #72211373, C2×#92838, … |
| maximal normal subgroup of | #72211373, C2×#92838, #72213174, … |
| maximal quotient of | #72210220, #72210671, #72210672, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34);