#18557

order	256
nilpotency class	3
isoclinism	isoclinic class of #18524
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

450 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#18557, #6348112, #6348114, …
maximal normal subgroup of	C₂×#18557, #6348112, #6348114, …
maximal quotient of	#165906, #165943, #166000, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 11545);