#6348189

order	512
nilpotency class	3
isoclinism	isoclinic class of #18562
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dh,f^b=fh,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283862, #72283889
maximal normal subgroup of	#72283862, #72283889

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255385);