#31077

order	256
nilpotency class	3
isoclinism	isoclinic class of #31077
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

536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^8|a^2=ef,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,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

34 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#31077⋊C₃, #72227427, #72227441, …
Fitting subgroup of	#31077⋊C₃
maximal subgroup of	#125651, #125887, #126286, …
maximal normal subgroup of	#125651, #125887, #126286, …
maximal quotient of	#343027, #343028, #343029, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 24065);