#359370

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

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

780 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:

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

Polycyclic presentation:

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

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, 266566);