#13069

order	256
nilpotency class	4
isoclinism	isoclinic class of #13068
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	#1893
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

225 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 73 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)(4,20,23,38)(5,24)(6,12)(7,30)(8,13)(9,17)(10,22,37,29)(11,26)(15,40)(16,47,50,63)(18,42)(19,44)(21,36,25,39)(27,31)(28,43)(32,59,62,64)(33,53)(34,45)(35,54)(41,49,60,51)(46,56,57,58)(48,55,52,61);(1,3,15,45,7,18,5,17,8,19,26,53,6,28,27,54)(2,9,31,42,12,34,11,14,13,35,24,43,30,33,40,44)(4,16,46,59,22,51,25,52,23,50,57,64,29,49,21,48)(10,32,58,63,38,61,39,41,37,62,56,47,20,55,36,60);(1,4)(2,10)(3,16)(5,25)(6,29)(7,22)(8,23)(9,32)(11,39)(12,38)(13,37)(14,41)(15,46)(17,52)(18,51)(19,50)(20,30)(21,27)(24,56)(26,57)(28,49)(31,58)(33,55)(34,61)(35,62)(36,40)(42,63)(43,47)(44,60)(45,59)(48,54)(53,64)

Polycyclic presentation:

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

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13069, #494651, #495900, …
maximal normal subgroup of	C₂×#13069, #494651, #495900, …
maximal quotient of	#137616, #137656, #137981, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6057);