#27698

order	256
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	Aut(G) = C₂¹¹×D₄ (order 16384) · Inn(G) = C₂²×D₄ · Out(G) = C₂⁹
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

355 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#27698, #6457054, #6457083, …
maximal normal subgroup of	C₂×#27698, #6457054, #6457083, …
maximal quotient of	#360350, #361363, #366377, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 20686);