#273548

order	512
nilpotency class	3
isoclinism	isoclinic class of #273435
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	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

861 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,d^2=h,e^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,g^c=ghi,e^d=ei,f^d=fhi

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, 180744);