#419839

order	512
nilpotency class	3
isoclinism	isoclinic class of #419839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1628 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=e,f^2=i,g^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,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^d=hi

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, 327035);