#47285

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

793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=h,b^2=fh,c^2=g,d^2=g,e^2=f,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×#47285, #7701412, #7701451, …
maximal normal subgroup of	C₂×#47285, #7701412, #7701451, …
maximal quotient of	#576803, #576852, #577109, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 40273);