#467807

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

860 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)(9,10,15,19)(11,20,18,14)(12,21,26,36)(13,17)(16,31,23,25)(22,29)(24,37)(27,38)(28,34)(30,35)(32,39)(33,40);(1,2,4,5)(3,6,7,8)(9,11,14,10,15,18,20,19)(12,23,25,36,26,16,31,21)(13,22,29,17,30,34,28,35)(24,38,39,33,40,32,27,37);(3,7)(6,8)(9,12,15,26)(10,16,19,23)(11,21,18,36)(13,27,30,39)(14,31,20,25)(17,33,35,37)(22,32,34,38)(24,29,40,28);(1,3,4,7)(2,6,5,8)(9,13,15,30)(10,17,19,35)(11,22,18,34)(12,24,26,40)(14,29,20,28)(16,32,23,38)(21,37,36,33)(25,39,31,27)

Polycyclic presentation:

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

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, 375003);