#405645

order	512
nilpotency class	3
isoclinism	isoclinic class of #394262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
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

1026 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312841);