#6684171

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	#3045	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 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)(17,18,22,21)(19,28,31,24)(20,34,36,44)(23,27,40,26)(25,39,37,38)(29,43,46,48)(30,32,42,33)(35,45,47,41);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,35)(21,28)(22,31)(23,32)(25,43)(26,42)(27,30)(29,39)(33,40)(34,41)(36,47)(37,48)(38,46)(44,45);(17,20,23,38,22,36,40,39)(18,25,26,34,21,37,27,44)(19,29,32,35,31,46,33,47)(24,41,42,48,28,45,30,43);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(18,21)(20,36)(24,28)(26,27)(29,46)(30,42)(35,47)(38,39);(3,11)(6,9)(10,16)(14,15)(17,21,22,18)(19,30,31,42)(20,37,36,25)(23,27,40,26)(24,33,28,32)(29,43,46,48)(34,38,44,39)(35,41,47,45)

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,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, 6591367);