#479802

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

732 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)(3,5)(4,6)(7,8)(9,10,14,19,15,18,13,20)(11,21,23,33,24,16,31,37)(12,25,29,17,28,35,30,36)(22,39,40,32,27,34,26,38);(1,3,4,7)(2,5,6,8)(9,11,14,31,15,24,13,23)(10,16,19,33,18,21,20,37)(12,26,29,27,28,40,30,22)(17,34,35,32,36,39,25,38);(9,12,13,30,15,28,14,29)(10,17,20,25,18,36,19,35)(11,22,23,40,24,27,31,26)(16,32,37,34,21,38,33,39);(2,6)(5,8)(9,11,15,24)(10,16,18,21)(12,27,28,22)(13,31,14,23)(17,32,36,38)(19,37,20,33)(25,39,35,34)(26,30,40,29)

Polycyclic presentation:

2^9|a^2=eg,b^2=efh,c^2=ef,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386998);