#358423

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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

994 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,g^b=gi,h^b=hi,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, 265619);