#18742

order	256
nilpotency class	3
isoclinism	isoclinic class of #18712
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

454 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)(3,7)(4,6)(5,8)(9,12)(10,15)(11,13)(14,16)(18,23)(19,26)(20,21)(25,29)(27,28)(30,31);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,18)(19,24)(20,23)(21,29)(22,25)(26,30)(27,31)(28,32);(1,3,5,10)(2,7,8,15)(4,11,12,16)(6,13,9,14)(17,19,22,28)(18,24,25,32)(20,26,21,27)(23,30,29,31);(2,8)(3,10)(6,9)(11,16)(17,20,22,21)(18,23,25,29)(19,27,28,26)(24,31,32,30)

Polycyclic presentation:

2^8|c^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,d^c=dg,e^c=eh,f^c=fh

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#18742, #6348357, #6348358, …
maximal normal subgroup of	C₂×#18742, #6348357, #6348358, …
maximal quotient of	#168143, #168178, #168225, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 11730);