#409542

order	512
nilpotency class	3
isoclinism	isoclinic class of #394590
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316738);