#10211

order	256
nilpotency class	3
isoclinism	isoclinic class of #10208
automorphism	Aut(G) order 12288
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

203 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,9,15)(3,16,22,30)(4,23,27,18)(5,28,33,51)(6,34,39,47)(7,17,41,53)(8,14,37,50)(10,42,20,24)(11,44,48,32)(12,38,52,64)(13,31,54,57)(19,55,46,63)(21,45,56,62)(25,61,40,35)(26,59,60,29)(36,58,49,43);(1,3,8,21)(2,4,14,26)(5,29,32,18)(6,35,38,19)(7,10,31,43)(9,22,37,56)(11,45,28,16)(12,49,47,24)(13,40,53,63)(15,27,50,60)(17,55,54,25)(20,57,58,41)(23,33,59,44)(30,48,62,51)(34,42,52,36)(39,61,64,46);(1,4,8,26)(2,10,14,43)(3,17,21,54)(5,30,32,62)(6,36,38,42)(7,40,31,63)(9,27,37,60)(11,46,28,61)(12,29,47,18)(13,22,53,56)(15,20,50,58)(16,44,45,33)(19,51,35,48)(23,52,59,34)(24,39,49,64)(25,57,55,41)

Polycyclic presentation:

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

Conjugacy classes & character table

25 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#10211⋊C₃
maximal subgroup of	C₂×#10211, #232819, #232828, …
maximal normal subgroup of	C₂×#10211, #232819, #232828, …
maximal quotient of	#102694, #102697, #102700, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 3199);