#323220

order	512
nilpotency class	3
isoclinism	isoclinic class of #323197
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19631	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

744 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=egh,c^2=h,d^2=ehi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,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, 230416);