#499544

order	512
nilpotency class	4
isoclinism	isoclinic class of #13398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

792 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=efh,d^2=i,e^2=h,f^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fh,h^b=hi,e^c=ehi,f^c=fhi,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, 406740);