#494686

order	512
nilpotency class	4
isoclinism	isoclinic class of #480460
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2098 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)(7,15,19,29)(8,20)(9,23,16,13)(10,14)(11,18)(12,21)(17,31,32,28)(22,27,30,35)(24,26)(25,36)(33,34);(5,7,16,18,10,19,9,24)(6,11,23,29,14,26,13,15)(8,17,30,33,21,32,22,36)(12,25,35,31,20,34,27,28);(2,4)(8,21)(12,20)(17,32)(22,30)(25,34)(27,35)(28,31)(33,36);(5,8,10,21)(6,12,14,20)(7,17,19,32)(9,22,16,30)(11,25,26,34)(13,27,23,35)(15,28,29,31)(18,33,24,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,g^b=gh,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, 401882);