#512585

order	512
nilpotency class	5
isoclinism	isoclinic class of #13569
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
socle	C₂	C₂²×AΣL(1,𝔽₁₆)
prime radical	C₂×#3018	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1480 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=ci,e^a=eg,g^a=gh,h^a=hi,e^b=ei,f^b=fg,g^b=gi,h^b=hi,f^e=fhi,g^e=gi,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279305, #72279311, #72279401, …
maximal normal subgroup of	#72279305, #72279311, #72279401, …
maximal quotient of	#72340590, #72340662

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419781);