#6544196

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533075
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2739 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6451392);