#6349730

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349684
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#62835	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2833 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256926);