#233832

order	512
nilpotency class	3
isoclinism	isoclinic class of #10218
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	#60940	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1352 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:

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

Polycyclic presentation:

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

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, 141028);