#41623

order	256
nilpotency class	2
isoclinism	isoclinic class of #40484
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	C₂⁴	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

683 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)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26)(27,33)(28,36)(29,30)(31,35)(32,38)(34,39)(37,40);(10,11)(15,17)(16,18)(22,23)(26,30)(31,33)(32,36)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,27,29,35)(26,31,30,33)(28,37,38,34)(32,40,36,39);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,29,38)(26,32,30,36)(27,34,35,37)(31,39,33,40);(1,3,4,7)(2,6,5,8)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,26)(27,31)(28,32)(29,30)(33,35)(34,39)(36,38)(37,40)

Polycyclic presentation:

2^8|a^2=h,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#41623, #7640116, #7640235, …
maximal normal subgroup of	C₂×#41623, #7640116, #7640235, …
maximal quotient of	#544789, #545236, #545869, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34611);