#237445

order	512
nilpotency class	3
isoclinism	isoclinic class of #237435
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1620 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)(3,4)(7,8)(9,10,15,20)(11,21,24,33)(12,25,26,34)(13,19,14,18)(16,28,31,23)(17,29,32,27)(22,30,36,40)(35,37,39,38);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,22)(13,28)(14,23)(15,24)(17,30)(18,33)(19,21)(20,31)(25,39)(26,36)(27,38)(29,37)(32,40)(34,35);(2,6)(3,4)(7,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,26)(16,30)(18,25)(19,34)(20,32)(21,35)(23,37)(24,36)(28,38)(31,40)(33,39);(12,26)(17,32)(22,36)(25,34)(27,29)(30,40)(35,39)(37,38)

Polycyclic presentation:

2^9|a^2=hi,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,e^b=eh,g^b=gi,d^c=dhi,e^c=ei,f^c=fhi,g^c=gi

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, 144641);