#394965

order	512
nilpotency class	3
isoclinism	isoclinic class of #394020
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1515 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)(3,6)(4,5)(7,8)(9,10,14,18)(11,15,20,30)(12,16,24,34)(13,25,28,17)(19,37,38,29)(21,35,26,31)(22,32,39,40)(23,36,27,33);(2,5)(6,8)(9,11)(10,15)(12,22)(13,26)(14,20)(16,32)(17,35)(18,30)(19,23)(21,28)(24,39)(25,31)(27,38)(29,33)(34,40)(36,37);(9,12)(10,16)(11,19)(13,27)(14,24)(15,29)(17,36)(18,34)(20,38)(21,39)(22,26)(23,28)(25,33)(30,37)(31,40)(32,35);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,19)(13,21)(14,20)(16,29)(17,31)(18,30)(22,23)(24,38)(25,35)(26,28)(27,39)(32,33)(34,37)(36,40)

Polycyclic presentation:

2^9|a^2=i,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,g^c=gi,h^c=hi,f^d=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, 302161);