#369043

order	512
nilpotency class	3
isoclinism	isoclinic class of #196299
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1858 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fg,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276239);