#46177

order	256
nilpotency class	2
isoclinism	isoclinic class of #45438
automorphism	Aut(G) = C₂¹⁵ (order 32768) · Inn(G) = C₂⁵ · Out(G) = C₂¹⁰
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂²×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

690 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^8|a^2=h,b^2=f,c^2=fg,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#46177, #7702796, #7702844, …
maximal normal subgroup of	C₂×#46177, #7702796, #7702844, …
maximal quotient of	#557284, #557305, #557490, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 39165);