#494704

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

2128 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)(5,6,9,14,10,13,23,24)(7,15,18,30,17,11,22,26)(8,19,21,33,20,12,27,28)(16,25,32,35,31,29,34,36);(5,7,9,22)(6,11,24,30)(8,16,21,34)(10,17,23,18)(12,25,33,36)(13,15,14,26)(19,29,28,35)(20,31,27,32);(2,4)(8,20)(12,19)(16,31)(21,27)(25,29)(28,33)(32,34)(35,36);(5,8,10,20)(6,12,13,19)(7,16,17,31)(9,21,23,27)(11,25,15,29)(14,28,24,33)(18,32,22,34)(26,35,30,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,g^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, 401900);