#464645

order	512
nilpotency class	3
isoclinism	isoclinic class of #454399
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60450	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1493 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371841);