#6441475

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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

1933 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)(18,23)(19,26)(25,31)(28,32);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,18,21,23)(19,27,26,29)(20,28,24,32)(22,31,30,25);(10,14)(13,16)(17,19)(18,24)(20,23)(21,26)(22,32)(25,29)(27,31)(28,30);(1,4,5,3)(2,6,7,8)(17,20,21,24)(18,19,23,26)(22,27,30,29)(25,32,31,28);(18,23)(19,26)(25,31)(28,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=fgi,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,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, 6348671);