#289114

order	512
nilpotency class	3
isoclinism	isoclinic class of #289075
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#546
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,6,8,4,7,5,3)(9,10,16,21)(11,22,12,18)(13,27,30,40)(14,31,15,20)(17,26,33,24)(19,29,35,28)(23,37,25,32)(34,38,36,39);(1,3,6,2,4,8,5,7)(9,11,15,26,16,12,14,24)(10,17,31,18,21,33,20,22)(13,23,29,39,30,25,28,38)(19,32,40,34,35,37,27,36);(1,2,4,7)(3,5,8,6)(9,12)(10,18)(11,16)(13,25)(14,26)(15,24)(17,20)(19,34)(21,22)(23,30)(27,32)(28,39)(29,38)(31,33)(35,36)(37,40);(9,13,15,29,16,30,14,28)(10,19,31,40,21,35,20,27)(11,23,26,39,12,25,24,38)(17,32,18,34,33,37,22,36)

Polycyclic presentation:

2^9|a^2=fhi,b^2=efh,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,g^b=gh,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196310);