#12954

order	256
nilpotency class	4
isoclinism	isoclinic class of #12952
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1864
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 190 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#12954, #494332, #495740, …
maximal normal subgroup of	C₂×#12954, #494332, #495740, …
maximal quotient of	#135907, #135931, #136224, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5942);