#17776

order	256
nilpotency class	3
isoclinism	isoclinic class of #17603
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₄²	C₂⁴
prime radical	#3508	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=f,b^2=g,c^2=g,d^2=g,e^2=h,f^2=gh,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#220661, #220663, #220669, …
maximal normal subgroup of	#220661, #220663, #220669, …
maximal quotient of	#163726, #163730, #163733, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10764);