#359374

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

Polycyclic presentation:

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

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, 266570);