#6355343

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355333
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	#60946	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1658 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)(7,13)(11,16)(12,15)(18,22)(19,25)(20,29)(24,31)(27,28)(30,32);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(17,18,20,22,21,24,29,31)(19,26,27,32,28,23,25,30);(3,11)(7,15)(10,16)(12,13)(17,19,21,28)(18,23,24,26)(20,27,29,25)(22,30,31,32);(18,24)(22,31)(23,26)(30,32);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(17,20,21,29)(18,22,24,31)(19,27,28,25)(23,30,26,32)

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,c^b=ch,d^b=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283862, #72283927, #72306859, …
maximal normal subgroup of	#72283862, #72283927, #72306859, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262539);