#6684168

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
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	#61337	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)(19,28)(20,25)(21,36)(22,26)(23,27)(24,32)(29,30)(31,43)(33,35)(34,41)(37,39)(38,42)(40,46)(44,45)(47,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,18)(19,29)(20,33)(21,26)(22,36)(23,27)(24,32)(25,35)(28,30)(31,40)(34,42)(37,39)(38,41)(43,46)(44,47)(45,48);(17,19,21,32,23,30,22,37)(18,24,26,28,27,39,36,29)(20,31,35,46,34,45,38,47)(25,40,42,43,41,48,33,44);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(19,30)(20,34)(24,39)(25,41)(28,29)(32,37)(33,42)(35,38);(3,11)(6,9)(10,16)(14,15)(17,20)(18,25)(19,31)(21,35)(22,38)(23,34)(24,40)(26,42)(27,41)(28,43)(29,44)(30,45)(32,46)(33,36)(37,47)(39,48)

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,e^d=ei,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, 6591364);