Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499867 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C2×#2202 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1106 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1106 conjugacy classes of subgroups — too many to draw. Only its 140 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 — 4 generators on 36 points:
(2,3)(5,6)(7,15)(8,11)(9,21)(10,24)(12,18)(13,19)(14,20)(16,34)(17,30)(22,27)(23,31)(25,33)(26,32)(28,35)(29,36);(1,2,4,3)(6,11)(7,16)(10,20)(12,28)(13,21)(15,33)(17,25)(18,36)(22,35)(23,26)(27,29)(30,34);(5,7)(6,12)(8,17)(9,22)(10,25)(11,27)(13,30)(14,28)(15,21)(16,20)(18,19)(23,36)(24,29)(26,35)(31,34)(32,33);(1,3,4,2)(6,11)(12,27)(13,21)(14,24)(15,30)(28,29)(31,32)(33,34)Polycyclic presentation:
2^9|b^2=ei,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72271412 |
| maximal normal subgroup of | #72271412 |
| maximal quotient of | #72271450 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407983);