#420657

order	512
nilpotency class	3
isoclinism	isoclinic class of #417665
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1018 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)(9,10,14,19,13,18,15,20)(11,21,24,36,23,16,30,32)(12,25,28,34,27,35,29,17)(22,38,26,31,39,33,40,37);(1,2,5,6)(3,7,4,8)(9,11,13,23)(10,16,18,21)(12,26,27,40)(14,24,15,30)(17,33,34,38)(19,32,20,36)(22,28,39,29)(25,37,35,31);(1,3,5,4)(2,7,6,8)(9,12,13,27)(10,17,18,34)(11,22,23,39)(14,29,15,28)(16,31,21,37)(19,35,20,25)(24,40,30,26)(32,38,36,33);(1,4,5,3)(2,7,6,8)(9,11)(10,16)(12,22)(13,23)(14,30)(15,24)(17,31)(18,21)(19,36)(20,32)(25,38)(26,29)(27,39)(28,40)(33,35)(34,37)

Polycyclic presentation:

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

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, 327853);