#6684564

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682658
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	#61217	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1621 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)(17,18,22,27)(19,28,32,29)(20,34,23,25)(21,26,37,41)(24,30,40,33)(31,43,47,44)(35,42,38,36)(39,45,46,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,18)(19,29)(20,25)(21,36)(22,27)(23,34)(24,33)(26,38)(28,32)(30,40)(31,46)(35,41)(37,42)(39,47)(43,45)(44,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,19,23,30,22,32,20,33)(18,24,34,29,27,40,25,28)(21,31,38,45,37,47,35,48)(26,39,42,44,41,46,36,43);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(17,20,22,23)(18,25,27,34)(19,30,32,33)(21,35,37,38)(24,29,40,28)(26,36,41,42)(31,45,47,48)(39,44,46,43);(3,13)(7,10)(11,12)(15,16)(17,21)(18,26)(19,31)(20,35)(22,37)(23,38)(24,39)(25,36)(27,41)(28,43)(29,44)(30,45)(32,47)(33,48)(34,42)(40,46)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=f,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,h^c=hi,h^e=hi

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, 6591760);