#60884

order	256
nilpotency class	3
isoclinism	isoclinic class of #60771
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	C₂×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

811 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#133274, #152131, #249875, …
maximal normal subgroup of	#133274, #152131, #249875, …
maximal quotient of	#6374396, #6374469, #6374979, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53872);