#421634

order	512
nilpotency class	3
isoclinism	isoclinic class of #418508
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

971 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328830);