#414859

order	512
nilpotency class	3
isoclinism	isoclinic class of #394602
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1063 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,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, 322055);