#419810

order	512
nilpotency class	3
isoclinism	isoclinic class of #419810
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,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, 327006);