#41580

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	C₂²×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

692 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:

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

Polycyclic presentation:

2^8|a^2=h,b^2=g,c^2=g,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×#41580, #7640071, #7640233, …
maximal normal subgroup of	C₂×#41580, #7640071, #7640233, …
maximal quotient of	#544681, #545192, #545697, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34568);