#462664

order	512
nilpotency class	3
isoclinism	isoclinic class of #458129
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

805 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=e,c^2=i,d^2=ef,e^2=i,g^2=i,h^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^c=de,e^c=ei,g^c=gi,h^c=hi,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, 369860);