#62322

order	256
nilpotency class	3
isoclinism	isoclinic class of #62307
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #525 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

897 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10,15,19)(11,16,22,31)(12,24,26,38)(13,18,14,28)(17,25,32,27)(20,33,36,40)(21,30,23,37)(29,34,39,35);(1,3)(2,5)(4,8)(6,7)(9,11,15,22)(10,16,19,31)(12,20,26,36)(13,21,14,23)(17,29,32,39)(18,30,28,37)(24,33,38,40)(25,34,27,35);(9,12,13,25,15,26,14,27)(10,17,28,38,19,32,18,24)(11,20,21,34,22,36,23,35)(16,29,37,40,31,39,30,33);(3,8)(5,7)(9,13,15,14)(10,18,19,28)(11,21,22,23)(12,25,26,27)(16,30,31,37)(17,24,32,38)(20,34,36,35)(29,33,39,40);(2,6)(5,7)(11,22)(12,26)(16,31)(17,32)(21,23)(24,38)(25,27)(30,37)

Polycyclic presentation:

2^8|a^2=h,b^2=h,c^2=gh,d^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,e^c=eh

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#62322, #10577613, #10577659, …
maximal normal subgroup of	C₂×#62322, #10577613, #10577659, …
maximal quotient of	#6386798, #6389160, #6389214, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55310);