#479737

order	512
nilpotency class	3
isoclinism	isoclinic class of #476609
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

752 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)(4,7)(5,8)(9,10,16,20,14,21,15,29)(11,12,18,30,23,17,32,25)(13,19,31,34,27,35,28,36)(22,24,37,40,38,26,33,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,23)(10,17)(12,21)(13,22,27,38)(15,18,16,32)(19,26)(20,30)(24,35)(25,29)(28,37,31,33)(34,40)(36,39);(1,3)(2,6)(4,8)(5,7)(9,12)(10,18)(11,20)(13,26)(14,17)(15,30)(16,25)(19,33)(21,32)(22,36)(23,29)(24,27)(28,39)(31,40)(34,38)(35,37);(3,8)(5,6)(9,13,14,27)(10,19,21,35)(11,22,23,38)(12,24,17,26)(15,31,16,28)(18,33,32,37)(20,36,29,34)(25,40,30,39)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386933);