#494376

order	512
nilpotency class	4
isoclinism	isoclinic class of #33551
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi

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, 401572);