#10691

order	256
nilpotency class	3
isoclinism	isoclinic class of #10690
automorphism	Aut(G) order 86016
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#401
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

186 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^8|a^2=defg,b^2=de,c^2=d,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fg,e^c=eg,f^c=fh

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#10691⋊C₇
Fitting subgroup of	#11709160, #10691⋊C₇
maximal subgroup of	C₂×#10691, #244495, #10690⋊C₂, …
maximal normal subgroup of	C₂×#10691, #244495, #10690⋊C₂, …
maximal quotient of	#121068, #146293, C₂×#10691, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 3679);