#460164

order	512
nilpotency class	3
isoclinism	isoclinic class of #454041
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

832 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:

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

Polycyclic presentation:

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

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, 367360);