#6350586

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350525
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#63001	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2710 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:

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

Polycyclic presentation:

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

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, 6257782);