#10612

order	256
nilpotency class	3
isoclinism	isoclinic class of #10611
automorphism	Aut(G) order 12288
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

164 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=de,b^2=dg,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,f^a=fg,c^b=cf,f^b=fh,d^c=dg,e^c=eh,f^c=fgh

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#10612⋊C₃
maximal subgroup of	C₂×#10612, #242363, #242365, …
maximal normal subgroup of	C₂×#10612, #242363, #242365, …
maximal quotient of	#105186, #105189, #105206, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 3600);