#13034

order	256
nilpotency class	4
isoclinism	isoclinic class of #13033
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1879
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

231 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 231 conjugacy classes of subgroups — too many to draw. Only its 67 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^8|a^2=f,b^2=dfg,c^2=h,d^2=g,e^2=g,f^2=g,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dg,e^b=egh,f^b=fg,g^b=gh,d^c=dgh,e^c=egh,f^c=fgh,g^c=gh

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13034, #494553, #495740, …
maximal normal subgroup of	C₂×#13034, #494553, #495740, …
maximal quotient of	#136702, #136704, #136736, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6022);