#6358573

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358426
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

1532 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)(17,18,23,27)(19,28,31,38)(20,25,22,35)(21,26,34,43)(24,36,40,29)(30,45,47,42)(32,41,37,44)(33,48,46,39);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(17,19,23,31)(18,24,27,40)(20,29,22,36)(21,33,34,46)(25,38,35,28)(26,42,43,45)(30,37,47,32)(39,41,48,44);(1,4)(2,8)(3,10)(5,14)(6,12)(7,15)(9,16)(11,13)(17,20,23,22)(18,25,27,35)(19,29,31,36)(21,32,34,37)(24,38,40,28)(26,41,43,44)(30,46,47,33)(39,42,48,45);(17,21,23,34)(18,26,27,43)(19,30,31,47)(20,32,22,37)(24,39,40,48)(25,41,35,44)(28,45,38,42)(29,46,36,33);(1,3,6,5)(2,7,9,13)(4,10,12,14)(8,15,16,11)(18,27)(24,40)(25,35)(26,43)(28,38)(39,48)(41,44)(42,45)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265769);