#6598515

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574100
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61159	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1732 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6505711);