#19382

order	256
nilpotency class	3
isoclinism	isoclinic class of #19368
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3251	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

297 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)(3,10)(4,8)(5,14)(6,9)(7,11)(12,15)(13,16)(17,18)(19,27)(20,26)(21,23)(22,25)(24,28)(29,32)(30,31);(1,3,5,11,6,12,4,13)(2,7,8,10,9,16,14,15)(18,23)(24,31)(25,26)(27,32);(1,2)(3,7)(4,14)(5,8)(6,9)(10,11)(12,16)(13,15)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,27)(26,32);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,27,31,32)

Polycyclic presentation:

2^8|b^2=eh,d^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,d^b=dg,d^c=dg,f^c=fh

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#19382, #6354072, #6354074, …
maximal normal subgroup of	C₂×#19382, #6354072, #6354074, …
maximal quotient of	#189131, #197832, #197837, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12370);