#414495

order	512
nilpotency class	3
isoclinism	isoclinic class of #401009
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

791 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,24)(13,18)(14,28)(15,19)(17,27)(20,29)(21,35)(22,31)(23,32)(25,39)(26,33)(30,34)(36,38)(37,40);(1,3,5,4)(2,6,7,8)(9,11,15,23)(10,16,19,32)(12,21,25,36)(13,20,14,22)(17,30,33,40)(18,29,28,31)(24,38,39,35)(26,34,27,37);(1,4,5,3)(2,6,7,8)(11,20)(13,14)(16,29)(18,28)(21,34)(22,23)(24,39)(26,27)(30,38)(31,32)(35,40)(36,37);(9,12)(10,17,19,33)(11,21)(13,26)(14,27)(15,25)(16,30,32,40)(18,24,28,39)(20,34)(22,37)(23,36)(29,38,31,35)

Polycyclic presentation:

2^9|b^2=efi,c^2=ef,d^2=gh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,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, 321691);