#6524890

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500157
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	#62835	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2861 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)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,38,22,26,21,27)(19,28,37,40,31,24,30,44)(20,32,35,25,34,42,36,43)(29,45,46,41,33,48,47,39);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,22,31)(18,24,26,28)(20,33,34,29)(21,37,23,30)(25,41,43,39)(27,44,38,40)(32,48,42,45)(35,46,36,47);(2,7)(3,9)(4,11)(12,16)(17,20)(18,25)(19,29)(21,35)(22,34)(23,36)(24,39)(26,43)(27,42)(28,41)(30,46)(31,33)(32,38)(37,47)(40,45)(44,48);(1,4,5,11)(2,8,7,14)(3,10,9,15)(6,12,13,16)(18,26)(20,34)(24,28)(27,38)(29,33)(35,36)(40,44)(46,47);(3,9)(6,13)(10,15)(12,16)(17,21,22,23)(18,27,26,38)(19,30,31,37)(20,35,34,36)(24,40,28,44)(25,42,43,32)(29,46,33,47)(39,45,41,48)

Polycyclic presentation:

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

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, 6432086);