#6621203

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1356 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)(3,6)(4,5)(7,8)(17,18)(19,28)(20,33)(21,25)(22,27)(23,32)(24,34)(26,29)(30,44)(31,38)(35,39)(36,45)(37,42)(40,43)(41,46)(47,48);(2,5)(6,8)(10,14)(12,15)(17,19,29,42,22,31,21,32)(18,23,25,38,27,37,26,28)(20,30,43,48,35,45,34,46)(24,36,39,47,40,44,33,41);(1,3,4,7)(2,6,5,8)(9,10)(11,12)(13,14)(15,16)(17,20,21,34,22,35,29,43)(18,24,26,33,27,40,25,39)(19,30,32,46,31,45,42,48)(23,36,28,41,37,44,38,47);(9,11,13,16)(10,15,14,12)(19,31)(23,37)(28,38)(30,45)(32,42)(36,44)(41,47)(46,48);(9,12,13,15)(10,11,14,16)(17,21,22,29)(18,25,27,26)(19,32,31,42)(20,34,35,43)(23,38,37,28)(24,39,40,33)(30,46,45,48)(36,47,44,41)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6528399);