#419849

order	512
nilpotency class	3
isoclinism	isoclinic class of #419450
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1652 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,f^c=fi,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, 327045);