#369044

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	#35329	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1922 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)(5,6,11,16,9,14,10,15)(7,17,20,29,19,12,26,28)(8,21,24,31,23,32,25,13)(18,30,35,33,34,36,22,27);(1,2,4,3)(5,7,10,26,9,19,11,20)(6,12,15,29,14,17,16,28)(8,22,25,34,23,35,24,18)(13,30,32,27,31,36,21,33);(5,8)(6,13,14,31)(7,18,19,34)(9,23)(10,24)(11,25)(12,27)(15,21,16,32)(17,33)(20,22,26,35)(28,36)(29,30);(1,3,4,2)(6,14)(12,17)(13,31)(15,16)(21,32)(27,33)(28,29)(30,36)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fg,d^2=ei,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, 276240);