#6684155

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682628
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	#3045	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1555 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)(17,18,22,25)(19,29,31,24)(20,33,32,44)(21,26,36,41)(23,28,39,27)(30,46,47,40)(34,42,35,48)(37,45,38,43);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,23)(21,35)(22,31)(25,29)(26,42)(27,33)(28,44)(30,37)(32,39)(34,36)(38,47)(40,45)(41,48)(43,46);(17,20,23,31,22,32,39,19)(18,24,27,33,25,29,28,44)(21,34,37,47,36,35,38,30)(26,40,43,42,41,46,45,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,25)(24,29)(26,41)(27,28)(33,44)(40,46)(42,48)(43,45);(3,11)(6,9)(10,16)(14,15)(17,21,22,36)(18,26,25,41)(19,30,31,47)(20,34,32,35)(23,37,39,38)(24,40,29,46)(27,43,28,45)(33,42,44,48)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=fi,e^2=i,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,h^c=hi,h^e=hi

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, 6591351);