#6464841

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372037);