#60581

order	256
nilpotency class	3
isoclinism	isoclinic class of #60372
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	#3005	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

1012 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)(3,6)(4,5)(7,8)(9,10,14,19)(11,15,22,29)(12,17,23,31)(13,16,25,18)(20,27,33,38)(21,26,35,28)(24,30,36,32)(34,37,40,39);(2,5)(6,8)(9,11,14,22)(10,15,19,29)(12,20,23,33)(13,21,25,35)(16,26,18,28)(17,27,31,38)(24,34,36,40)(30,37,32,39);(10,16)(13,25)(15,26)(17,30)(18,19)(21,35)(24,36)(27,37)(28,29)(31,32)(34,40)(38,39);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,20)(13,24)(14,23)(15,27)(16,30)(18,32)(19,31)(21,34)(22,33)(25,36)(26,37)(28,39)(29,38)(35,40);(12,23)(17,31)(20,33)(24,36)(27,38)(30,32)(34,40)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6577388, #6577389, #6577400, …
maximal normal subgroup of	#6577388, #6577389, #6577400, …
maximal quotient of	#6371020, #6371813, #6371815, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53569);