#13006

order	256
nilpotency class	4
isoclinism	isoclinic class of #13005
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	#1874
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

236 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 73 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13006, #494480, #495738, …
maximal normal subgroup of	C₂×#13006, #494480, #495738, …
maximal quotient of	#136615, #136643, #136887, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5994);