#6456912

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1512 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)(9,10)(11,14)(12,13)(15,16)(18,24)(25,32)(26,29)(27,28);(1,3,5,4)(2,6,8,7)(17,18,22,24)(19,29,30,26)(20,25,31,32)(21,28,23,27);(1,3,5,4)(2,7,8,6)(10,13)(14,16)(17,19)(18,25)(20,21)(22,30)(23,31)(24,32)(26,28)(27,29);(1,4,5,3)(2,6,8,7)(17,20,22,31)(18,26,24,29)(19,21,30,23)(25,28,32,27);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,24,28)(19,20,30,31)(25,29,32,26)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fi,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364108);