#6351987

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351885
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	#63059	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2614 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259183);