#46994

order	256
nilpotency class	2
isoclinism	isoclinic class of #46350
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

676 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)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(25,26,30,34)(27,31,35,40)(28,32,38,37)(29,33,39,36);(1,3,5,4)(2,6,8,7)(10,12)(11,18)(13,21)(22,23)(27,35)(29,39)(31,40)(33,36);(1,2)(3,6)(4,7)(5,8)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,27)(26,31)(28,33)(29,37)(30,35)(32,39)(34,40)(36,38);(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,28)(26,32)(27,36)(29,31)(30,38)(33,35)(34,37)(39,40);(1,4,5,3)(2,7,8,6)(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,29)(26,33)(27,37)(28,31)(30,39)(32,35)(34,36)(38,40)

Polycyclic presentation:

2^8|a^2=fgh,b^2=fg,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#46994, #7700720, #7700776, …
maximal normal subgroup of	C₂×#46994, #7700720, #7700776, …
maximal quotient of	#576842, #576874, #577103, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 39982);