#368790

order	512
nilpotency class	3
isoclinism	isoclinic class of #196143
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1714 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)(5,6,10,15,11,14,9,24)(7,12,19,27,17,28,18,33)(8,20,22,30,23,13,25,31)(16,26,21,35,32,36,34,29);(2,4)(5,7,10,19,11,17,9,18)(6,12,15,27,14,28,24,33)(8,21,22,32,23,34,25,16)(13,29,31,26,20,35,30,36);(5,8,9,22,11,23,10,25)(6,13,24,31,14,20,15,30)(7,16,19,34,17,32,18,21)(12,26,27,29,28,36,33,35);(2,4)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,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, 275986);