#6435890

order	512
nilpotency class	3
isoclinism	isoclinic class of #6435890
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2644
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1173 conjugacy classes of subgroups; only the 488 normal subgroups are shown.

normal subgroups only

This group has 1173 conjugacy classes of subgroups — too many to draw. Only its 488 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343086);