#420703

order	512
nilpotency class	3
isoclinism	isoclinic class of #417661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1166 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327899);