#469144

order	512
nilpotency class	3
isoclinism	isoclinic class of #455828
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

861 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:

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

Polycyclic presentation:

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

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, 376340);