#427431

order	512
nilpotency class	3
isoclinism	isoclinic class of #418535
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1226 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 334627);