#6454851

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437963
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62788	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2985 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6362047);