#6464920

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=h,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,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, 6372116);