#499540

order	512
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#476
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

806 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=efh,d^2=i,e^2=h,f^2=hi,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,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, 406736);