#499337

order	512
nilpotency class	4
isoclinism	isoclinic class of #13331
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#459
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

790 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)(5,6,12,20)(7,21,26,28)(8,23,13,25)(9,15,31,19)(10,32,34,17)(11,18,22,16)(14,33,36,29)(24,27,35,30);(5,7,22,35,9,14,34,25,12,26,11,24,31,36,10,23)(6,13,32,29,19,27,18,28,20,8,17,33,15,30,16,21);(2,4)(5,8)(6,14)(7,19)(9,27)(10,33)(11,21)(12,13)(15,26)(16,25)(17,24)(18,23)(20,36)(22,28)(29,34)(30,31)(32,35);(5,9,12,31)(6,15,20,19)(7,14,26,36)(8,27,13,30)(10,22,34,11)(16,17,18,32)(21,33,28,29)(23,35,25,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406533);