#380185

order	512
nilpotency class	3
isoclinism	isoclinic class of #219518
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

986 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:

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

Polycyclic presentation:

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

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, 287381);