#6362713

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362510
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#61434	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1529 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269909);