#273710

order	512
nilpotency class	3
isoclinism	isoclinic class of #273691
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	#62088	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

884 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 180906);