#494361

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	C₄	C₂×B(4,𝔽₂)
commutator	C₂×D₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
socle	C₂	C₂×#1867
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236884, #72237544, #72237549, …
maximal normal subgroup of	#72236884, #72237544, #72237549, …
maximal quotient of	#72303729, C₂×#494361, #72312636

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401557);