#41531

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	C₂×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

717 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)(9,10,14,15)(11,18,16,12)(13,17,21,22)(19,24,23,20)(25,26)(27,34)(28,29)(30,31)(32,37)(33,39)(35,36)(38,40);(1,3,5,4)(2,6,8,7)(10,15)(11,16)(17,22)(19,23)(26,31)(29,33)(32,34)(36,40);(1,2)(3,6)(4,7)(5,8)(9,11,14,16)(10,12,15,18)(13,19,21,23)(17,20,22,24)(25,27,30,37)(26,32,31,34)(28,38,39,35)(29,36,33,40);(1,4,5,3)(2,7,8,6)(9,12)(10,16)(11,15)(13,20)(14,18)(17,23)(19,22)(21,24)(25,28)(26,33)(27,35)(29,31)(30,39)(32,36)(34,40)(37,38);(9,13,14,21)(10,17,15,22)(11,19,16,23)(12,20,18,24)(25,29,30,33)(26,28,31,39)(27,36,37,40)(32,38,34,35)

Polycyclic presentation:

2^8|a^2=fgh,b^2=fg,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×#41531, #7640222, #7640232, …
maximal normal subgroup of	C₂×#41531, #7640222, #7640232, …
maximal quotient of	#544723, #545210, #545722, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34519);