#362385

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2283
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

888 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)(7,16)(8,20,22,12)(9,13)(10,15)(11,18)(14,23)(17,30,32,29)(19,25)(21,28,35,27)(24,31,34,36)(26,33);(1,2,3,4)(5,7,9,18,10,19,23,33)(6,11,14,16,15,26,13,25)(8,17,21,31,22,32,35,36)(12,24,28,29,20,34,27,30);(2,4)(5,8)(6,12)(7,17)(9,21)(10,22)(11,24)(13,27)(14,28)(15,20)(16,29)(18,31)(19,32)(23,35)(25,30)(26,34)(33,36);(6,13,15,14)(11,25,26,16)(12,27,20,28)(24,30,34,29)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269581);