#6593559

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#24237	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6500755);