#453821

order	512
nilpotency class	3
isoclinism	isoclinic class of #421159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

659 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:

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

Polycyclic presentation:

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

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, 361017);