#416211

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

1314 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:

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

Polycyclic presentation:

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

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, 323407);