#406062

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

867 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)(7,19,24,34)(8,25,15,9)(10,16)(11,17)(12,18)(13,22,32,23)(14,27,28,29)(20,30)(21,36)(26,35)(31,33);(1,2)(3,4)(5,7)(6,13,18,32)(8,26,15,31)(9,21)(10,22)(11,23)(12,24)(14,33)(16,19,17,34)(20,27,36,29)(25,30)(28,35);(2,4)(5,8)(6,14)(7,20)(9,17)(10,27)(11,29)(12,15)(13,30)(16,25)(18,28)(19,35)(21,32)(22,26)(23,31)(24,36)(33,34);(5,9,10,14,12,25,11,28)(6,15,16,29,18,8,17,27)(7,21,22,33,24,30,23,35)(13,31,19,20,32,26,34,36)

Polycyclic presentation:

2^9|a^2=ef,b^2=h,d^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, 313258);