#445426

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

752 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=e,c^2=g,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 352622);