#6359223

order	512
nilpotency class	3
isoclinism	isoclinic class of #6359112
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1550 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)(3,7)(4,11)(5,13)(6,9)(8,10)(12,15)(14,16)(18,24)(21,22)(25,38)(26,39)(27,36)(28,41)(30,35)(32,42)(33,37)(40,44)(43,48)(45,46);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,18)(19,28)(20,32)(21,24)(22,36)(23,27)(25,31)(26,33)(29,44)(30,38)(34,39)(35,41)(37,42)(40,45)(43,46)(47,48);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(17,19)(18,25)(20,29)(21,35)(22,30)(23,31)(24,38)(26,40)(27,28)(32,48)(33,46)(34,47)(36,41)(37,45)(39,44)(42,43);(17,20,23,34)(18,26,27,42)(19,29,31,47)(21,33,22,37)(24,39,36,32)(25,40,28,43)(30,45,35,46)(38,44,41,48);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,21,23,22)(18,24,27,36)(19,30,31,35)(20,33,34,37)(25,41,28,38)(26,39,42,32)(29,45,47,46)(40,48,43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266419);