#318614

order	512
nilpotency class	3
isoclinism	isoclinic class of #318601
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

563 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=i,d^2=e,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dhi,f^c=fi,g^c=gi

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, 225810);