#6496559

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470421
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	#60568	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1648 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)(19,27)(20,32)(21,37)(22,25)(23,30)(24,35)(26,38)(28,39)(29,44)(31,45)(33,40)(34,41)(36,42)(43,47)(46,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19,21,30,22,31,38,44)(18,23,26,27,25,29,37,45)(20,33,34,47,36,28,35,46)(24,40,42,48,41,39,32,43);(3,9)(6,13)(10,15)(14,16)(17,20)(18,24)(19,28)(21,34)(22,36)(23,39)(25,41)(26,42)(27,43)(29,40)(30,46)(31,33)(32,37)(35,38)(44,47)(45,48);(3,9)(6,13)(10,15)(14,16)(17,18)(19,29)(20,24)(21,26)(22,25)(23,31)(27,44)(28,40)(30,45)(32,35)(33,39)(34,42)(36,41)(37,38)(43,47)(46,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(18,25)(23,29)(24,41)(26,37)(27,45)(32,42)(39,40)(43,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,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,g^c=gi,e^d=ei

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, 6403755);