#478975

order	512
nilpotency class	3
isoclinism	isoclinic class of #476293
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

585 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386171);