#31076

order	256
nilpotency class	3
isoclinism	isoclinic class of #31076
automorphism	Aut(G) = #72340865 (order 12288) · Inn(G) = H(3,𝔽₄) · Out(G) = D₄×S₄
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
socle	C₂	#2679
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

546 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 4 generators on 16 points:

(2,8,10,9)(3,11,13,16);(1,2)(3,12)(4,11)(5,9)(6,8)(7,10)(13,14)(15,16);(1,3)(2,4)(5,11)(6,16)(7,13)(8,12)(9,14)(10,15);(1,4)(2,3)(5,14)(6,12)(7,15)(8,16)(9,11)(10,13)

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#31076⋊C₃, C₂×#31076⋊C₃, #72267788
Fitting subgroup of	#31076⋊C₃
maximal subgroup of	#125652, #125653, #125890, …
maximal normal subgroup of	#125652, #125653, #125890, …
maximal quotient of	#343003, #343004, #343005, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 24064);