#12542

order	256
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	Aut(G) = C₄×#72224760 (order 6144) · Inn(G) = C₄² · Out(G) = C₄×GL(2,ℤ/4)
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	C₄²
commutator	C₄	C₄³
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1726
prime radical	C₂³	#131
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

175 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,5,10,6,11,21,32,7,12,22,33,23,34,45,56)(3,8,14,25,16,27,38,49,15,26,37,48,39,50,58,62)(4,9,18,29,20,31,43,54,19,30,42,53,44,55,60,64)(13,24,35,46,17,28,40,51,36,47,57,61,41,52,59,63);(1,3)(2,8)(4,17,19,41)(5,14)(6,16)(7,15)(9,28,30,52)(10,25)(11,27)(12,26)(13,20,36,44)(18,40,42,59)(21,38)(22,37)(23,39)(24,31,47,55)(29,51,53,63)(32,49)(33,48)(34,50)(35,43,57,60)(45,58)(46,54,61,64)(56,62);(1,4)(2,9)(3,13,15,36)(5,18)(6,20)(7,19)(8,24,26,47)(10,29)(11,31)(12,30)(14,35,37,57)(16,17,39,41)(21,43)(22,42)(23,44)(25,46,48,61)(27,28,50,52)(32,54)(33,53)(34,55)(38,40,58,59)(45,60)(49,51,62,63)(56,64)

Polycyclic presentation:

2^8|a^2=d,b^2=e,c^2=f,d^2=g,g^2=h,c^b=cg,f^b=fh,e^c=eh

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#12542⋊C₃
maximal subgroup of	#141373, #141374, #141375, …
maximal normal subgroup of	#141373, #141374, #141375, …
maximal quotient of	#126529, #126533, #126536, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5530);