#60983

order	256
nilpotency class	3
isoclinism	isoclinic class of #60756
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
prime radical	#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,28,37)(20,30,32,40)(21,33,35,25)(24,34,39,31)(27,41,42,48)(29,45,44,38)(36,46,47,43);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,22,28)(18,23,26,37)(20,27,32,42)(21,29,35,44)(24,36,39,47)(25,38,33,45)(30,41,40,48)(31,43,34,46);(3,10)(6,9)(11,16)(13,15)(17,20,22,32)(18,24,26,39)(19,27,28,42)(21,34,35,31)(23,36,37,47)(25,40,33,30)(29,46,44,43)(38,48,45,41);(2,7)(6,9)(8,14)(13,15);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(19,28)(23,37)(27,42)(29,44)(36,47)(38,45)(41,48)(43,46)

Polycyclic presentation:

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

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6352778, #6352779, #6352788, …
maximal normal subgroup of	#6352778, #6352779, #6352788, …
maximal quotient of	#6374725, #6374729, #6375095, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53971);