#6441437

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1928 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348633);