#6356859

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356733
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#3483	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1513 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)(3,12)(4,10)(5,11)(6,8)(7,15)(9,13)(14,16)(17,18,23,27)(19,28,32,45)(20,25,35,43)(21,26,22,37)(24,30,39,31)(29,44,33,48)(34,41,36,42)(38,47,40,46);(2,8)(9,16)(10,12)(11,15)(17,19,22,31,23,32,21,30)(18,24,26,28,27,39,37,45)(20,33,34,47,35,29,36,46)(25,40,42,48,43,38,41,44);(1,3,6,4)(2,9,8,16)(5,13,7,14)(10,15,12,11);(1,4,6,3)(2,10,8,12)(5,13,7,14)(9,15,16,11)(17,20)(18,25)(19,29)(21,36)(22,34)(23,35)(24,38)(26,42)(27,43)(28,44)(30,47)(31,46)(32,33)(37,41)(39,40)(45,48);(1,5,6,7)(2,11,8,15)(3,13,4,14)(9,10,16,12)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,34,35,36)(24,28,39,45)(25,41,43,42)(29,46,33,47)(38,48,40,44)

Polycyclic presentation:

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

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, 6264055);