#23768

order	256
nilpotency class	3
isoclinism	isoclinic class of #23387
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	C₂³	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	C₂⁴	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

347 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)(19,22)(20,24)(21,23)(25,29)(26,30)(27,31)(28,32);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(19,25)(20,27)(22,29)(24,31)(26,28)(30,32);(10,14)(13,16)(17,19,21,26)(18,22,23,30)(20,28,27,25)(24,32,31,29);(1,3,4,7)(2,6,5,8)(18,23)(22,30)(24,31)(29,32)

Polycyclic presentation:

2^8|b^2=f,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gh,g^c=gh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#23768, #6432468, #6432477, …
maximal normal subgroup of	C₂×#23768, #6432468, #6432477, …
maximal quotient of	#358139, #358388, #358391, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 16756);