#94521

order	512
nilpotency class	6
isoclinism	isoclinic class of #94516
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7338
commutator	C₂³×C₄	C₂×C₈
Frattini, cosocle	#1784	C₂²
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1784 ◁ G
LECS	G ▷ #1784 ▷ C₂³×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁

282 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 24 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=i,c^2=i,d^2=f,f^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ci,d^b=dei,f^b=fh,d^c=dg,f^c=fi,e^d=eh,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1717);