#6351208

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351101
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	#62913	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2694 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258404);