#499386

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	C₂⁵	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

716 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=e,d^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fh,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, 406582);