#6605392

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	#2428	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1582 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6512588);