#17309

order	256
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	Aut(G) order 23592960
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

733 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:

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

Polycyclic presentation:

2^8|a^2=h,b^2=g,c^2=g,d^2=h,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	#11709340, #11709341, #17309⋊C₅, …
Fitting subgroup of	#11709340, #11709341, #11709342, …
maximal subgroup of	#125870, #126013, #144629, …
maximal normal subgroup of	#125870, #126013, #144629, …
maximal quotient of	#156673, #160357, #160416, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10297);