#427948

order	512
nilpotency class	3
isoclinism	isoclinic class of #417728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

1283 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)(3,4)(6,12)(7,17,20,19)(8,21,23,22)(13,26,24,28)(14,29,25,30)(15,16)(18,32)(31,34);(5,6,9,12)(7,13,20,24)(8,14,23,25)(10,16,11,15)(17,26,19,28)(18,33,32,27)(21,29,22,30)(31,36,34,35);(5,7)(6,13)(8,18)(9,20)(10,19)(11,17)(12,24)(14,27)(15,26)(16,28)(21,34)(22,31)(23,32)(25,33)(29,36)(30,35);(2,4)(5,8,11,22,9,23,10,21)(6,14,16,29,12,25,15,30)(7,18,17,31,20,32,19,34)(13,27,28,36,24,33,26,35)

Polycyclic presentation:

2^9|a^2=ei,b^2=i,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,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, 335144);