#472331

order	512
nilpotency class	3
isoclinism	isoclinic class of #465214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

864 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)(3,6)(4,5)(7,8)(9,10,16,20)(11,21,24,36)(12,25,29,30)(13,14,19,15)(17,32,34,23)(18,31,27,28)(22,35,26,33)(37,38,40,39);(2,5)(6,8)(9,11,14,23,16,24,15,32)(10,17,13,21,20,34,19,36)(12,26,31,38,29,22,28,39)(18,35,30,40,27,33,25,37);(3,7)(6,8)(9,12,15,28,16,29,14,31)(10,18,19,25,20,27,13,30)(11,22,23,39,24,26,32,38)(17,33,21,37,34,35,36,40);(1,3,4,7)(2,6,5,8)(9,13,14,20,16,19,15,10)(11,21,23,34,24,36,32,17)(12,27,28,30,29,18,31,25)(22,35,38,37,26,33,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379527);