#467623

order	512
nilpotency class	3
isoclinism	isoclinic class of #455828
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
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

706 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=ei,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,h^a=hi,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, 374819);