#16243

order	256
nilpotency class	2
isoclinism	isoclinic class of #15257
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

495 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)(5,8)(9,10)(11,13)(12,15)(14,16)(17,18,20,21)(19,23,24,22)(25,26,29,28)(27,31,32,30);(10,11)(12,16)(18,21)(22,23)(25,27,29,32)(26,30,28,31);(1,2,4,6)(3,5,7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,22)(20,24)(21,23)(25,27)(26,31)(28,30)(29,32);(1,3)(2,5)(4,7)(6,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,22)(20,21)(23,24)(25,28,29,26)(27,30,32,31)

Polycyclic presentation:

2^8|a^2=fg,b^2=ef,c^2=egh,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#16243⋊C₃, #72238088
Fitting subgroup of	#16243⋊C₃
maximal subgroup of	#125832, #126184, C₂×#16243, …
maximal normal subgroup of	#125832, #126184, C₂×#16243, …
maximal quotient of	#154973, #155503, #155538, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 9231);