#476655

order	512
nilpotency class	3
isoclinism	isoclinic class of #476654
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1056 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383851);