#415877

order	512
nilpotency class	3
isoclinism	isoclinic class of #394271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60415	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1610 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,22)(13,26)(14,17)(16,25)(18,28)(19,34)(20,36)(21,30)(23,39)(24,33)(27,32)(29,35)(31,38)(37,40);(2,5)(6,7)(9,11,14,21)(10,15,17,30)(12,23)(13,20,28,38)(16,29)(18,31,26,36)(19,24)(22,34)(25,37)(27,35)(32,40)(33,39);(9,12)(10,16)(11,19)(13,27)(14,24)(15,29)(17,32)(18,33)(20,37)(21,23)(22,26)(25,28)(30,40)(31,39)(34,36)(35,38);(1,3,4,8)(2,6,5,7)(10,17)(15,30)(16,32)(18,26)(22,33)(29,40)(31,36)(34,39)

Polycyclic presentation:

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

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, 323073);