#17054

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

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

501 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)(11,13)(12,15)(14,16)(17,18,21,22)(19,23,24,20)(26,27)(28,32);(2,5)(6,8)(10,11)(14,15)(18,22)(19,24)(25,26,29,27)(28,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,14,16,15)(17,19)(18,23)(20,22)(21,24)(25,27,29,26)(28,30,32,31);(1,3,4,7)(2,6,5,8)(9,12,13,16)(10,14,11,15)(17,20,21,23)(18,24,22,19)(25,28)(26,30)(27,31)(29,32)

Polycyclic presentation:

2^8|a^2=ef,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	#17054⋊C₃, #72260950
Fitting subgroup of	#17054⋊C₃
maximal subgroup of	#104783, #104784, C₂×#17054, …
maximal normal subgroup of	#104783, #104784, C₂×#17054, …
maximal quotient of	#155791, #156096, #156228, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10042);