#470974

order	512
nilpotency class	3
isoclinism	isoclinic class of #461251
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

635 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)(3,5)(4,6)(7,8)(9,10,15,19)(11,20)(12,23)(13,17,29,34)(14,18,25,24)(16,31)(21,35)(22,30)(26,36)(27,39)(28,33,38,37)(32,40);(1,3,4,7)(2,5,6,8)(9,11,15,23)(10,16)(12,24)(13,21,29,36)(14,22,25,31)(17,27)(18,20)(19,30)(26,37)(28,32,38,39)(33,35)(34,40);(9,12,25,16,15,20,14,30)(10,11,18,31,19,23,24,22)(13,27,38,35,29,40,28,26)(17,32,33,21,34,39,37,36);(2,6)(5,8)(9,13,14,28,15,29,25,38)(10,17,18,33,19,34,24,37)(11,21,22,32,23,36,31,39)(12,26,16,27,20,35,30,40)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=ef,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=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, 378170);