#7024155

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#59049	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1639 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6931351);