#376654

order	512
nilpotency class	3
isoclinism	isoclinic class of #24985
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

855 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=eh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^c=fi,g^c=gi

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, 283850);