#43626

order	256
nilpotency class	2
isoclinism	isoclinic class of #42996
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

760 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#43626, #7667788, #7667975, …
maximal normal subgroup of	C₂×#43626, #7667788, #7667975, …
maximal quotient of	#535542, #535607, #536174, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 36614);