#479759

order	512
nilpotency class	3
isoclinism	isoclinic class of #476613
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eh,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,g^c=gi,e^d=ei,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, 386955);