#243029

order	512
nilpotency class	3
isoclinism	isoclinic class of #243027
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

728 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150225);