#6815750

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814811
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	#61216	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1651 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,23,29)(19,24,34,45)(20,32,39,46)(21,26,41,28)(22,27,42,30)(25,36,47,38)(31,37,33,44)(35,40,48,43);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,24)(20,35)(21,31)(22,25)(23,34)(26,37)(27,36)(28,44)(29,45)(30,38)(32,40)(33,41)(39,48)(42,47)(43,46);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20,23,39)(18,25)(19,30)(21,38,41,36)(22,37,42,44)(24,43,45,40)(26,46)(27,34)(28,32)(29,47)(31,48)(33,35);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(17,21,23,41)(18,26,29,28)(19,31,34,33)(20,36,39,38)(22,40,42,43)(24,37,45,44)(25,32,47,46)(27,48,30,35);(3,13)(7,10)(11,12)(15,16)(17,22,21,40,23,42,41,43)(18,27,26,48,29,30,28,35)(19,32,33,25,34,46,31,47)(20,37,38,24,39,44,36,45)

Polycyclic presentation:

2^9|a^2=i,c^2=fgh,d^2=fi,e^2=h,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

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, 6722946);