#451034

order	512
nilpotency class	3
isoclinism	isoclinic class of #417544
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1302 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)(3,6)(4,5)(7,8)(9,10,14,30,15,19,33,20)(11,21,25,35,24,16,31,37)(12,26,27,34,23,17,32,38)(13,28,29,18,22,36,40,39);(2,5)(3,7)(9,11)(10,16)(12,22)(13,23)(14,31)(15,24)(17,36)(18,38)(19,21)(20,37)(25,33)(26,28)(27,29)(30,35)(32,40)(34,39);(9,12,15,23)(10,17,19,26)(11,22,24,13)(14,32,33,27)(16,28,21,36)(18,37,39,35)(20,38,30,34)(25,29,31,40);(1,3,4,7)(2,6,5,8)(9,13,15,22)(10,18)(11,23,24,12)(14,29,33,40)(16,34)(17,37)(19,39)(20,28)(21,38)(25,32,31,27)(26,35)(30,36)

Polycyclic presentation:

2^9|a^2=gh,c^2=i,d^2=ei,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^b=de,f^b=fi,h^b=hi,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, 358230);