#494740

order	512
nilpotency class	4
isoclinism	isoclinic class of #480482
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#35676	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2042 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)(5,6,11,14,12,16,9,21)(7,17,15,27,22,10,20,18)(8,13,23,30,25,31,26,28)(19,32,29,36,34,24,35,33);(5,7,18,6,12,22,27,16)(8,19,33,13,25,34,36,31)(9,20,17,14,11,15,10,21)(23,29,24,28,26,35,32,30);(2,4)(5,8)(6,13)(7,19)(9,26)(10,24)(11,23)(12,25)(14,30)(15,29)(16,31)(17,32)(18,33)(20,35)(21,28)(22,34)(27,36);(5,9,12,11)(6,14,16,21)(7,20,22,15)(8,23,25,26)(10,18,17,27)(13,28,31,30)(19,29,34,35)(24,36,32,33)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,d^c=di,g^e=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, 401936);