#422104

order	512
nilpotency class	3
isoclinism	isoclinic class of #419041
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1043 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,f^c=fi,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, 329300);