#46774

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

740 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:

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

Polycyclic presentation:

2^8|a^2=f,b^2=fg,c^2=f,d^2=fg,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×#46774, #7700435, #7700519, …
maximal normal subgroup of	C₂×#46774, #7700435, #7700519, …
maximal quotient of	#576910, #576977, #577110, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 39762);