#62282

order	256
nilpotency class	3
isoclinism	isoclinic class of #62265
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

908 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)(11,16)(12,24)(13,18)(14,29)(15,19)(17,27)(20,36)(21,31)(22,32)(23,33)(25,39)(26,35)(28,40)(30,38)(34,37);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,25)(13,21)(14,22)(15,23)(17,34)(18,31)(19,33)(20,28)(24,39)(26,38)(27,37)(29,32)(30,35)(36,40);(9,12,14,27,15,28,13,26)(10,17,18,24,19,35,29,40)(11,20,22,38,23,25,21,37)(16,30,31,36,33,34,32,39);(3,8)(5,7)(9,13,15,14)(10,18,19,29)(11,21,23,22)(12,26,28,27)(16,31,33,32)(17,24,35,40)(20,37,25,38)(30,36,34,39);(2,6)(5,7)(9,13,15,14)(10,18,19,29)(11,22,23,21)(12,26,28,27)(16,32,33,31)(17,24,35,40)(20,38,25,37)(30,39,34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#62282, #10577582, #10577663, …
maximal normal subgroup of	C₂×#62282, #10577582, #10577663, …
maximal quotient of	#6386755, #6389148, #6389149, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55270);