#403904

order	512
nilpotency class	3
isoclinism	isoclinic class of #394747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

872 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)(5,6,10,15)(7,16)(8,13,21,29)(9,12,23,14)(11,19)(17,30)(18,26)(20,24)(22,27,34,28)(25,33)(31,35)(32,36);(5,7,9,18,10,20,23,19)(6,11,14,24,15,26,12,16)(8,17,22,32,21,31,34,33)(13,25,28,35,29,36,27,30);(6,12,15,14)(8,21)(11,24,26,16)(13,28,29,27)(17,31)(22,34)(25,30,36,35)(32,33);(2,4)(5,8,9,22,10,21,23,34)(6,13,12,27,15,29,14,28)(7,17,18,32,20,31,19,33)(11,25,16,30,26,36,24,35)

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,c^2=g,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di

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, 311100);