#19309

order	256
nilpotency class	3
isoclinism	isoclinic class of #19304
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	#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

300 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#19309, #6353968, #6353992, …
maximal normal subgroup of	C₂×#19309, #6353968, #6353992, …
maximal quotient of	#189123, #197808, #197810, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12297);