#359277

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

982 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266473);