#409968

order	512
nilpotency class	3
isoclinism	isoclinic class of #393979
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1581 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,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, 317164);