#6496555

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470415
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1650 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)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,18)(19,29)(20,34)(21,26)(22,28)(23,27)(24,33)(25,36)(30,42)(31,41)(32,40)(35,39)(37,43)(38,44)(45,47)(46,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19,21,33,23,32,22,31)(18,24,28,29,27,41,26,40)(20,35,36,46,37,30,38,47)(25,42,34,48,44,39,43,45);(3,9)(6,13)(10,15)(14,16)(17,20)(18,25)(19,30)(21,36)(22,38)(23,37)(24,39)(26,43)(27,44)(28,34)(29,45)(31,46)(32,35)(33,47)(40,48)(41,42);(3,9)(6,13)(10,15)(14,16)(17,21,23,22)(18,26,27,28)(19,31,32,33)(20,36,37,38)(24,29,41,40)(25,43,44,34)(30,46,35,47)(39,45,42,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,27)(19,32)(25,44)(26,28)(30,35)(31,33)(34,43)(46,47)

Polycyclic presentation:

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

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, 6403751);