#60206

order	256
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	Aut(G) order 16384
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂⁴×C₄
Frattini, cosocle	C₂×C₄	C₂⁵
prime radical	#3455	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

939 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)(9,10,13,18,14,17,24,30)(11,15,21,27,20,26,33,37)(12,16,22,29,23,28,34,38)(19,25,32,36,31,35,39,40);(2,5)(6,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,25)(17,26)(18,27)(22,32)(23,31)(24,33)(28,35)(29,36)(30,37)(34,39)(38,40);(9,12,13,22,14,23,24,34)(10,16,18,29,17,28,30,38)(11,19,21,32,20,31,33,39)(15,25,27,36,26,35,37,40);(1,3,4,7)(2,6,5,8)(11,20)(15,26)(19,31)(21,33)(25,35)(27,37)(32,39)(36,40);(10,17)(15,26)(16,28)(18,30)(25,35)(27,37)(29,38)(36,40)

Polycyclic presentation:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#129547, #209064, #209065, …
maximal normal subgroup of	#129547, #209064, #209065, …
maximal quotient of	#6369982, #6369987, #6370027, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53194);