#13339

order	256
nilpotency class	4
isoclinism	isoclinic class of #13307
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

267 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 61 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(5,6,12,18)(7,19,24,54)(8,25,31,44)(9,32,37,50)(10,17,11,16)(13,38,42,21)(14,27,47,28)(15,35,51,36)(20,39,58,67)(22,48,23,53)(26,63,57,43)(29,49,30,62)(33,40,60,41)(34,45,65,46)(52,55,68,56)(59,61,64,66);(1,2)(3,4)(5,7,12,24)(6,13,18,42)(8,26,31,57)(9,33,37,60)(10,23,11,22)(14,43,47,63)(15,48,51,53)(16,41,17,40)(19,50,54,32)(20,30,58,29)(21,35,38,36)(25,61,44,66)(27,55,28,56)(34,64,65,59)(39,46,67,45)(49,68,62,52);(2,4)(5,8)(6,14)(7,20)(9,34)(10,30)(11,29)(12,31)(13,39)(15,49)(16,46)(17,45)(18,47)(19,52)(21,59)(22,57)(23,26)(24,58)(25,32)(27,35)(28,36)(33,56)(37,65)(38,64)(40,63)(41,43)(42,67)(44,50)(48,61)(51,62)(53,66)(54,68)(55,60)

Polycyclic presentation:

2^8|a^2=h,b^2=h,d^2=gh,e^2=gh,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fh,g^a=gh,c^b=cf,d^b=dg,e^b=eg,f^b=fh,g^b=gh,d^c=dgh,e^c=eg,f^c=fh,g^c=gh

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13339, #498877, #498879, …
maximal normal subgroup of	C₂×#13339, #498877, #498879, …
maximal quotient of	#142638, #142639, #142647, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6327);