#60345

order	256
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄	C₂⁵
commutator, socle	C₂²	C₂⁴×C₄
prime radical	#3482	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

872 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,25,22,20,24,29)(19,23,27,30,26,28,31,32);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,23)(20,28)(21,27)(22,26)(24,31)(25,30)(29,32);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,26)(23,28)(27,31)(30,32);(2,8)(6,9)(7,16)(14,15)(19,26)(23,28)(27,31)(30,32);(17,20,21,29,22,18,24,25)(19,23,27,30,26,28,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#60345⋊C₃
maximal subgroup of	#129677, #210206, #210207, …
maximal normal subgroup of	#129677, #210206, #210207, …
maximal quotient of	#6370510, #6370515, #6370594, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53333);