#17053

order	256
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	Aut(G) order 393216
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

565 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^8|a^2=f,b^2=gh,c^2=f,d^2=g,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#17053⋊C₃, C₂×#17053⋊C₃
Fitting subgroup of	#17053⋊C₃
maximal subgroup of	#104777, #104778, C₂×#17053, …
maximal normal subgroup of	#104777, #104778, C₂×#17053, …
maximal quotient of	#155791, #156097, #156229, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10041);