#6350601

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

2703 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(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)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(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)(2,8,9,13)(3,11,12,14)(7,15,16,10)(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);(2,9)(7,16)(8,13)(10,15)(17,21)(18,26)(19,31)(20,35)(22,38)(23,39)(24,37)(25,40)(27,42)(28,43)(29,36)(30,44)(32,46)(33,47)(34,45)(41,48);(21,37)(26,36)(31,35)(38,39)(40,48)(42,43)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^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, 6257797);