#364985

order	512
nilpotency class	3
isoclinism	isoclinic class of #23906
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60415	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1664 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)(5,6,10,14,11,15,9,22)(7,12,17,28,18,27,23,26)(8,19,20,13,21,31,24,30)(16,32,34,29,35,36,33,25);(1,2,4,3)(5,7,10,17,11,18,9,23)(6,12,14,28,15,27,22,26)(8,16,20,34,21,35,24,33)(13,29,31,36,30,25,19,32);(5,8)(6,13)(7,16)(9,24)(10,20)(11,21)(12,25)(14,31)(15,30)(17,34)(18,35)(19,22)(23,33)(26,36)(27,29)(28,32);(1,3,4,2)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=e,f^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,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272181);