#405848

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

971 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313044);