Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13673 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator | #127 | C22×C4 |
| Frattini, prime radical, cosocle | C2×#127 | C23 |
| LCS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
521 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 521 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6)(7,11)(8,20,21,12)(9,13,24,14)(10,15)(16,25,31,34)(17,30,18,26)(19,27)(22,29,23,28)(32,35,33,36);(1,2,3,4)(5,7,10,19)(6,11,15,27)(8,16,23,33)(9,18,24,17)(12,25,29,36)(13,30,14,26)(20,34,28,35)(21,31,22,32);(5,8,10,23)(6,12,15,29)(7,16,19,33)(9,21,24,22)(11,25,27,36)(13,28,14,20)(17,32,18,31)(26,34,30,35)Polycyclic presentation:
2^9|a^2=f,b^2=dfi,c^2=i,e^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eghi,g^a=gh,h^a=hi,e^b=eghi,h^b=hi,d^c=dg,e^c=eh,f^c=fg,g^c=gi,e^d=eh,g^d=gi,f^e=fh,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72279456, #72279943, #72280012, … |
| maximal normal subgroup of | #72279456, #72279943, #72280012, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59779);