#6355496

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355483
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1647 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)(5,14)(6,9)(7,11)(12,15)(13,16)(18,23)(19,26)(20,21)(25,29)(27,28)(30,32);(1,3,5,11,6,12,4,13)(2,7,8,10,9,16,14,15)(17,18,21,23,22,25,20,29)(19,24,27,30,28,31,26,32);(17,19,22,28)(18,24,25,31)(20,26,21,27)(23,30,29,32);(18,25)(23,29)(24,31)(30,32);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,16,10)(17,20,22,21)(18,23,25,29)(19,26,28,27)(24,30,31,32)

Polycyclic presentation:

2^9|b^2=f,c^2=i,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,d^b=di,e^b=eh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283748
maximal normal subgroup of	#72283748

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262692);