#17404

order	256
nilpotency class	3
isoclinism	isoclinic class of #17392
automorphism	Aut(G) order 32768
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

480 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)(3,11,14,7)(4,8)(5,9)(6,10)(12,16,13,15)(17,18,21,24,22,25,20,29)(19,23,28,31,26,30,27,32);(1,3,6,14)(2,7,10,11)(4,12,5,13)(8,15,9,16)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,15,11,16)(19,26)(23,30)(27,28)(31,32);(1,5,6,4)(2,9,10,8)(3,13,14,12)(7,16,11,15)(17,20,22,21)(18,24,25,29)(19,27,26,28)(23,31,30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#105764, #106097, #106104, …
maximal normal subgroup of	#105764, #106097, #106104, …
maximal quotient of	#162611, #162617, #162623, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10392);