#419566

order	512
nilpotency class	3
isoclinism	isoclinic class of #417665
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1081 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,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, 326762);