#451029

order	512
nilpotency class	3
isoclinism	isoclinic class of #417571
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

1295 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)(9,10,15,31,16,20,14,21)(11,22,25,33,12,17,24,18)(13,27,28,19,29,37,30,36)(23,34,38,35,26,40,39,32);(2,5)(3,7)(9,11,16,12)(10,17,20,22)(13,23,29,26)(14,25,15,24)(18,31,33,21)(19,35,36,32)(27,40,37,34)(28,39,30,38);(9,12,16,11)(10,18,20,33)(13,26,29,23)(14,24,15,25)(17,21,22,31)(19,34,36,40)(27,35,37,32)(28,38,30,39);(1,3,4,7)(2,6,5,8)(9,13)(10,19,20,36)(11,23)(12,26)(14,30)(15,28)(16,29)(17,32,22,35)(18,34,33,40)(21,27,31,37)(24,39)(25,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,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, 358225);