#453928

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

659 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)(11,15)(12,23,25,36)(13,16)(14,18)(17,26,33,28)(19,29)(20,37,24,40)(21,30)(22,32)(27,35)(31,38,34,39);(1,3,5,8)(2,6,4,7)(9,11,19,22,13,21,14,27)(10,15,29,32,16,30,18,35)(12,24,26,38,25,20,28,39)(17,34,23,37,33,31,36,40);(1,2,5,4)(3,6,8,7)(10,16)(14,19)(15,30)(17,33)(22,27)(26,28)(31,34)(38,39);(1,4,5,2)(3,6,8,7)(9,12)(10,17)(11,20,21,24)(13,25)(14,28)(15,31,30,34)(16,33)(18,36)(19,26)(22,39,27,38)(23,29)(32,40,35,37)

Polycyclic presentation:

2^9|a^2=e,b^2=egh,c^2=efi,d^2=efg,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,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, 361124);