#17085

order	256
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	Aut(G) order 786432
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

697 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 20 points:

(2,4)(6,8)(13,14,18,20)(15,16,17,19);(6,8)(10,12)(13,15,18,17)(14,19,20,16);(5,6)(7,8)(9,10)(11,12)(13,16,18,19)(14,15,20,17);(1,2)(3,4)(5,6)(7,8)(13,17,18,15)(14,16,20,19)

Polycyclic presentation:

2^8|a^2=e,b^2=e,c^2=e,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#17085⋊C₃, #72227425, #72227439, …
Fitting subgroup of	#17085⋊C₃
maximal subgroup of	#125657, #125806, #125883, …
maximal normal subgroup of	#125657, #125806, #125883, …
maximal quotient of	#156471, #156660, #156663, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10073);