#504797

order	512
nilpotency class	4
isoclinism	isoclinic class of #504384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1771 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

552 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=efi,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,e^b=eh,f^b=fhi,h^b=hi,e^c=ehi,f^c=fh,h^c=hi,e^d=ehi,f^d=fh,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322361
maximal normal subgroup of	#72322361

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411993);