#420149

order	512
nilpotency class	3
isoclinism	isoclinic class of #417704
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1141 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=e,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, 327345);