#146496

order	512
nilpotency class	4
isoclinism	isoclinic class of #12721
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1782
commutator	C₂²×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂	#10726
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

829 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,e^b=ei,f^b=fh,d^c=di,f^c=fi,f^d=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, 53692);