#464957

order	512
nilpotency class	3
isoclinism	isoclinic class of #455116
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23439	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1141 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)(11,20)(12,15)(13,26)(14,18)(16,22)(17,30)(19,25)(21,36)(23,33)(24,31)(27,28)(29,34)(32,38)(35,39)(37,40);(2,5)(6,8)(9,11,15,23,14,22,25,31)(10,16,12,24,18,20,19,33)(13,21,29,38,28,37,30,39)(17,32,26,40,34,35,27,36);(1,3)(2,6)(4,7)(5,8)(9,12,25,18,14,19,15,10)(11,16,31,24,22,20,23,33)(13,27,30,17,28,26,29,34)(21,35,39,36,37,32,38,40);(3,7)(6,8)(9,13)(10,17)(11,21)(12,26)(14,28)(15,29)(16,32)(18,34)(19,27)(20,35)(22,37)(23,38)(24,40)(25,30)(31,39)(33,36)

Polycyclic presentation:

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

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, 372153);