#16248

order	256
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	Aut(G) order 786432
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

625 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:

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

Polycyclic presentation:

2^8|a^2=ef,b^2=efh,c^2=gh,d^2=efg,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	#16248⋊C₃, #72266642, C₂×#16248⋊C₃, …
Fitting subgroup of	#16248⋊C₃, #72266668
maximal subgroup of	#125837, #125869, #125952, …
maximal normal subgroup of	#125837, #125869, #125952, …
maximal quotient of	#155021, #155402, #155619, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 9236);