#6354414

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354359
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	#3480	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1521 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(17,18,22,26)(19,27,29,40)(20,24,32,38)(21,34,28,25)(23,35,36,31)(30,41,43,48)(33,45,42,39)(37,46,47,44);(17,19,28,35,22,29,21,31)(18,23,34,40,26,36,25,27)(20,30,42,46,32,43,33,44)(24,37,45,48,38,47,39,41);(1,3,6,12)(2,7,8,15)(4,11,5,14)(9,16,13,10)(19,29)(23,36)(27,40)(30,43)(31,35)(37,47)(41,48)(44,46);(2,8)(7,15)(9,13)(10,16)(17,20,22,32)(18,24,26,38)(19,30,29,43)(21,33,28,42)(23,37,36,47)(25,39,34,45)(27,41,40,48)(31,44,35,46);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(20,32)(24,38)(30,43)(33,42)(37,47)(39,45)(41,48)(44,46)

Polycyclic presentation:

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

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, 6261610);