#243020

order	512
nilpotency class	3
isoclinism	isoclinic class of #10637
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#397
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

731 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=ei,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,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, 150216);