#474780

order	512
nilpotency class	3
isoclinism	isoclinic class of #469776
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

888 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:

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

Polycyclic presentation:

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

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, 381976);