#63040

order	256
nilpotency class	2
isoclinism	isoclinic class of #63011
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

4254 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 16 points:

(1,2,6,5)(3,8,4,7)(9,10,14,13)(11,16,12,15);(1,2,6,5)(3,8,4,7)(9,11,14,12)(10,15,13,16);(1,3,6,4)(2,7,5,8)(9,11,14,12)(10,15,13,16);(9,11,14,12)(10,15,13,16);(1,4,6,3)(2,7,5,8)(9,12,14,11)(10,15,13,16);(1,5,6,2)(3,8,4,7)(9,13,14,10)(11,16,12,15)

Polycyclic presentation:

2^8|a^2=h,b^2=h,c^2=h,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11710846, #72272545
Fitting subgroup of	#11710844, #11710845, #11710846, …
maximal subgroup of	#346946, #347205, #493660, …
maximal normal subgroup of	#346946, #347205, #493660, …
maximal quotient of	#7627258, #7627335, #7627395, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 56028);