#6526826

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500489
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	#3455	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1607 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)(3,6,9,12)(4,7,11,14)(10,13,15,16)(17,18,24,28)(19,29,34,42)(20,35,37,39)(21,26,38,36)(22,27,23,40)(25,33,43,32)(30,31,45,48)(41,47,46,44);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19)(18,25)(20,30)(21,31)(22,32)(23,33)(24,34)(26,41)(27,42)(28,43)(29,40)(35,47)(36,46)(37,45)(38,48)(39,44);(3,9)(4,11)(6,12)(7,14)(17,20,23,36,24,37,22,26)(18,21,27,35,28,38,40,39)(19,30,33,46,34,45,32,41)(25,31,42,47,43,48,29,44);(1,4,5,11)(2,7,8,14)(3,10,9,15)(6,13,12,16)(17,21)(18,26)(19,31)(20,27)(22,39)(23,35)(24,38)(25,41)(28,36)(29,45)(30,42)(32,44)(33,47)(34,48)(37,40)(43,46);(3,9)(6,12)(10,15)(13,16)(17,22,24,23)(18,27,28,40)(19,32,34,33)(20,26,37,36)(21,35,38,39)(25,42,43,29)(30,41,45,46)(31,47,48,44)

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,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, 6434022);