#41528

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

674 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:

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

Polycyclic presentation:

2^8|a^2=gh,b^2=g,c^2=f,d^2=fg,e^2=f,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×#41528, #7640116, #7640232, …
maximal normal subgroup of	C₂×#41528, #7640116, #7640232, …
maximal quotient of	#544715, #545210, #545709, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34516);