#453842

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

721 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361038);