#12751

order	256
nilpotency class	4
isoclinism	isoclinic class of #12750
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1786
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#578	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

438 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 85 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)(3,14,18,42)(4,20,23,38)(5,25)(6,11)(7,13)(8,16,32,48)(9,22,34,24)(10,27)(12,29)(15,43)(17,41,19,35)(21,53,56,33)(26,39,49,31)(28,36)(30,58)(37,47)(40,50)(44,59)(45,63)(46,62)(51,64)(52,55,61,57)(54,60);(1,3)(2,8)(4,15)(5,26)(6,17)(7,18)(9,30)(10,35)(11,31)(12,39)(13,32)(14,36)(16,47)(19,29)(20,51)(21,54)(22,45)(23,46)(24,50)(25,42)(27,48)(28,49)(33,43)(34,60)(37,41)(38,59)(40,61)(44,55)(52,63)(53,62)(56,58)(57,64);(1,4,7,23)(2,9,13,34)(3,15,18,46)(5,21,28,56)(6,24,29,22)(8,30,32,60)(10,33,37,53)(11,38,12,20)(14,40,42,63)(16,44,48,64)(17,50,19,45)(25,52,36,61)(26,54,49,58)(27,57,47,55)(31,59,39,51)(35,43,41,62)

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#12751, #493556, #495740, …
maximal normal subgroup of	C₂×#12751, #493556, #495740, …
maximal quotient of	#129523, #129566, #129790, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5739);