#6544037

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
prime radical	#62962	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2761 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)(19,29,32,39)(20,33,36,42)(21,27)(22,26)(23,28)(24,31,40,37)(25,35,41,34)(30,38)(43,44)(45,47)(46,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,13)(11,16)(12,14)(17,19,23,32)(18,24,28,40)(20,30,36,46)(21,31,22,37)(25,38,41,48)(26,39,27,29)(33,47,42,43)(34,44,35,45);(2,6)(7,13)(8,14)(15,16)(17,20)(18,25)(19,30)(21,34)(22,35)(23,36)(24,38)(26,33)(27,42)(28,41)(29,43)(31,44)(32,46)(37,45)(39,47)(40,48);(1,3,5,10)(2,7,6,13)(4,11,12,9)(8,16,14,15)(17,21,23,22)(18,26,28,27)(19,31,32,37)(20,34,36,35)(24,39,40,29)(25,33,41,42)(30,44,46,45)(38,47,48,43);(1,4,5,12)(2,8,6,14)(3,9,10,11)(7,15,13,16)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,f^b=fi,g^c=gi,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, 6451233);