#415235

order	512
nilpotency class	3
isoclinism	isoclinic class of #396836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

664 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ef,c^2=e,d^2=hi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,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, 322431);