#6362409

order	512
nilpotency class	3
isoclinism	isoclinic class of #6362318
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	#61898	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1552 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)(3,7)(4,11)(5,13)(6,9)(8,10)(12,15)(14,16)(17,18,22,25)(19,27,28,43)(20,24,33,41)(21,34,37,26)(23,35,38,30)(29,45,46,40)(31,48,47,39)(32,44,36,42);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,19,22,28)(18,23,25,38)(20,31,33,47)(21,35,37,30)(24,40,41,45)(26,43,34,27)(29,36,46,32)(39,44,48,42);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11)(19,28)(23,38)(27,43)(29,46)(30,35)(31,47)(39,48)(40,45);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,20,22,33)(18,24,25,41)(19,29,28,46)(21,36,37,32)(23,39,38,48)(26,44,34,42)(27,45,43,40)(30,47,35,31);(2,9)(7,13)(8,16)(11,15)(18,25)(23,38)(24,41)(26,34)(27,43)(39,48)(40,45)(42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6269605);