#494720

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×D₄	C₂⁴
prime radical	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2024 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)(7,15,18,10)(8,12)(9,13)(11,14)(16,29,32,20)(17,30,34,23)(19,25)(21,26)(22,27)(24,28)(31,36,33,35);(5,7,15,14)(6,11,18,10)(8,16,29,26)(9,17,30,28)(12,21,32,20)(13,24,34,23)(19,31,36,27)(22,33,35,25);(2,4)(5,8,11,21)(6,12,14,26)(7,16,18,32)(9,22,24,19)(10,20,15,29)(13,27,28,25)(17,33,34,31)(23,36,30,35);(5,9,11,24)(6,13,14,28)(7,17,18,34)(8,19,21,22)(10,23,15,30)(12,25,26,27)(16,31,32,33)(20,35,29,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,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, 401916);