#13248

order	256
nilpotency class	4
isoclinism	isoclinic class of #13247
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1972
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

197 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 197 conjugacy classes of subgroups — too many to draw. Only its 65 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)(4,22,26,52)(5,28)(6,11)(7,32)(8,14)(9,18)(10,25,39,46)(12,30)(13,29)(16,41)(17,33,49,62)(19,35)(20,42)(21,45)(23,53,55,37)(24,56,58,40)(27,60,57,38)(31,44)(34,50)(36,51)(43,48,63,54)(47,61,59,64);(1,3,16,19,7,20,5,18,8,21,6,31,29,50,30,51)(2,9,12,42,13,35,11,15,14,36,28,34,32,44,41,45)(4,23,25,59,27,49,58,48,26,55,46,47,57,17,24,54)(10,37,52,63,40,62,60,64,39,53,22,43,56,33,38,61);(1,4,6,25,7,27,30,58,8,26,16,46,29,57,5,24)(2,10,12,22,13,40,11,38,14,39,28,52,32,56,41,60)(3,17,19,48,20,23,18,47,21,49,31,54,50,55,51,59)(9,33,34,61,35,37,45,63,36,62,42,64,44,53,15,43)

Polycyclic presentation:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13248, #495237, #495900, …
maximal normal subgroup of	C₂×#13248, #495237, #495900, …
maximal quotient of	#140746, #140748, #140782, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6236);