#368735

order	512
nilpotency class	3
isoclinism	isoclinic class of #196152
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

1724 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)(5,6,11,15,10,16,9,14)(7,12,18,26,19,27,23,30)(8,20,21,29,22,31,24,13)(17,32,34,25,35,36,33,28);(2,4)(5,7,11,18,10,19,9,23)(6,12,15,26,16,27,14,30)(8,17,21,34,22,35,24,33)(13,28,20,32,29,25,31,36);(5,8,11,24,10,22,9,21)(6,13,14,20,16,29,15,31)(7,17,18,33,19,35,23,34)(12,25,30,36,27,28,26,32);(2,4)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gi,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, 275931);