#6482191

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470804
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1592 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)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18)(19,28)(20,33)(21,26)(22,37)(23,27)(24,31)(25,36)(29,43)(30,39)(32,42)(34,48)(35,40)(38,44)(41,46)(45,47);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19,22,31,23,32,21,30)(18,24,26,28,27,39,37,42)(20,29,35,45,36,46,34,44)(25,38,40,41,33,47,48,43);(3,9)(6,13)(10,11)(14,15)(17,20,22,35,23,36,21,34)(18,25,26,40,27,33,37,48)(19,29,31,45,32,46,30,44)(24,38,28,41,39,47,42,43);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15);(2,8)(6,13)(7,16)(14,15)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,34,36,35)(24,28,39,42)(25,40,33,48)(29,44,46,45)(38,41,47,43)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6389387);