#450136

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1046 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)(5,6,11,16)(7,9,18,26)(8,19,24,32)(10,27,29,15)(12,28,25,14)(13,22,33,23)(17,30,20,35)(21,31,36,34);(5,7,11,18)(6,12,16,25)(8,20)(9,15,26,27)(10,28,29,14)(13,30)(17,24)(19,21)(22,31)(23,34)(32,36)(33,35);(5,8)(6,13)(7,17)(9,21)(10,22)(11,24)(12,30)(14,31)(15,32)(16,33)(18,20)(19,27)(23,29)(25,35)(26,36)(28,34);(2,4)(5,9,10,25,11,26,29,12)(6,14,15,18,16,28,27,7)(8,21,22,35,24,36,23,30)(13,31,32,20,33,34,19,17)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357332);