#6362407

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	#3520	C₂²
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) = 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)(19,27,31,38)(20,24,34,41)(21,35,36,26)(23,29,39,30)(28,43,32,46)(33,47,48,42)(37,44,40,45);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,19,22,31)(18,23,25,39)(20,32,34,28)(21,29,36,30)(24,40,41,37)(26,38,35,27)(33,45,48,44)(42,43,47,46);(1,4,6,12)(2,8,9,16)(3,10,5,14)(7,15,13,11)(17,20,22,34)(18,24,25,41)(19,28,31,32)(21,33,36,48)(23,37,39,40)(26,42,35,47)(27,43,38,46)(29,44,30,45);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(19,29)(21,36)(23,38)(26,35)(27,39)(28,44)(30,31)(32,45)(33,48)(37,46)(40,43)(42,47);(2,9)(7,13)(8,16)(11,15)(18,25)(23,39)(24,41)(26,35)(27,38)(37,40)(42,47)(43,46)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=i,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, 6269603);