#6350993

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350909
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	#60837	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2608 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:

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

Polycyclic presentation:

2^9|a^2=i,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,d^c=dh,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, 6258189);