#499422

order	512
nilpotency class	4
isoclinism	isoclinic class of #13398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#481
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

602 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:

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

Polycyclic presentation:

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

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, 406618);