#420247

order	512
nilpotency class	3
isoclinism	isoclinic class of #417704
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

1065 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=ei,f^2=i,g^2=i,h^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^d=hi

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, 327443);