#494761

order	512
nilpotency class	4
isoclinism	isoclinic class of #494749
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1898
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#60369	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#585 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2005 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237529, #72237542, #72237549, …
maximal normal subgroup of	#72237529, #72237542, #72237549, …
maximal quotient of	#72312051, #72312834, C₂×#494761

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401957);