#29187

order	256
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	Aut(G) = C₂¹¹×D₄ (order 16384) · Inn(G) = C₂²×D₄ · Out(G) = C₂⁹
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

299 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=g,b^2=g,c^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#29187, #6463861, #6463870, …
maximal normal subgroup of	C₂×#29187, #6463861, #6463870, …
maximal quotient of	#362186, #363521, #365498, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 22175);