#17081

order	256
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	Aut(G) order 786432
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)(5,8)(9,10,12,14)(11,15,16,13)(18,21)(19,23)(25,26)(27,29)(28,30)(31,32);(17,18)(19,22)(20,21)(23,24)(25,26,28,30)(27,29,32,31);(1,2,4,6)(3,5,7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,21)(22,23)(25,26)(27,31)(28,30)(29,32);(1,3,4,7)(2,5,6,8)(9,11,12,16)(10,13,14,15)(17,19,20,23)(18,22,21,24)(25,27)(26,29)(28,32)(30,31)

Polycyclic presentation:

2^8|a^2=efg,b^2=g,c^2=fh,d^2=h,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	#17081⋊C₃, #72238087
Fitting subgroup of	#17081⋊C₃
maximal subgroup of	#104725, #104726, #104727, …
maximal normal subgroup of	#104725, #104726, #104727, …
maximal quotient of	#155868, #156460, #158527, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 10069);