#17338

order	256
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	Aut(G) order 98304
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

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

356 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:

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

Polycyclic presentation:

2^8|a^2=e,b^2=f,e^2=g,f^2=h,c^a=cg,d^a=dh,c^b=cgh,d^b=dg

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#17338⋊C₃, #72235180, #72235181, …
Fitting subgroup of	#17338⋊C₃
maximal subgroup of	#109732, #109737, #109739, …
maximal normal subgroup of	#109732, #109737, #109739, …
maximal quotient of	#180830, #180857, #181814, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10326);