#6684160

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61329	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1579 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6591356);