#464867

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1023 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=e,d^2=e,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, 372063);