#409541

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

Polycyclic presentation:

2^9|a^2=eg,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, 316737);