#13243

order	256
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	Aut(G) = #72316158 (order 4096) · Inn(G) = #466 · Out(G) = C₂⁴×C₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#466
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1970
prime radical	C₂×#508	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

205 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 69 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)(3,15,20,39)(4,22)(5,12)(6,29)(7,31)(8,14)(9,18,34,27)(10,24)(11,16)(13,30)(17,32,47,58)(19,35,43,57)(21,33,49,42)(23,51)(25,36)(26,53)(28,40)(37,55)(38,44)(41,61,60,46)(45,62,54,50)(48,64,63,59)(52,56);(1,3,16,43,30,49,5,27,8,20,6,19,7,21,28,18)(2,9,12,33,31,57,11,39,14,34,40,42,13,35,29,15)(4,17,44,61,56,63,23,54,26,47,24,46,25,48,55,45)(10,32,22,50,51,64,36,60,38,58,53,62,37,59,52,41);(1,4,5,23,7,25,16,44,8,26,28,55,30,56,6,24)(2,10,29,52,13,37,40,53,14,38,11,36,31,51,12,22)(3,17,27,54,21,48,43,61,20,47,18,45,49,63,19,46)(9,32,15,41,35,59,42,62,34,58,39,60,57,64,33,50)

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#152530, #152533, #152542, …
maximal normal subgroup of	#152530, #152533, #152542, …
maximal quotient of	#140728, #140729, #140730, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6231);