#243912

order	512
nilpotency class	3
isoclinism	isoclinic class of #232771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

795 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=gh,d^c=dh,e^c=ei,f^c=fi,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151108);