#6685508

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
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	#60769	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1768 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)(11,21)(12,17)(13,18)(14,19)(15,20)(16,23)(22,25)(24,35)(26,27)(28,34)(29,32)(30,33)(31,37)(36,39)(38,40);(2,5)(3,7)(9,10)(11,22)(12,19)(13,27)(14,17)(15,20)(16,23)(18,26)(21,25)(24,31)(28,33)(29,32)(30,34)(35,37)(36,38)(39,40);(9,11,12,23,15,25,14,29)(10,16,19,21,20,32,17,22)(13,24,26,37,28,39,30,38)(18,31,33,35,34,40,27,36);(2,5)(6,8)(9,12,15,14)(10,17,20,19)(11,23,25,29)(13,26,28,30)(16,22,32,21)(18,27,34,33)(24,37,39,38)(31,36,40,35);(1,3,4,7)(2,6,5,8)(9,13)(10,18)(11,24)(12,26)(14,30)(15,28)(16,31)(17,27)(19,33)(20,34)(21,35)(22,36)(23,37)(25,39)(29,38)(32,40)

Polycyclic presentation:

2^9|c^2=g,d^2=i,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,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, 6592704);