#62518

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 ◁ #529 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

827 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=fh,c^2=g,d^2=h,e^2=f,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×#62518, #10579826, #10579842, …
maximal normal subgroup of	C₂×#62518, #10579826, #10579842, …
maximal quotient of	#6386902, #6389533, #6389555, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55506);