#13031

order	256
nilpotency class	4
isoclinism	isoclinic class of #13030
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

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

normal subgroups only

This group has 228 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13031, #494550, #495738, …
maximal normal subgroup of	C₂×#13031, #494550, #495738, …
maximal quotient of	#136699, #136733, #136908, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6019);