#94232

order	512
nilpotency class	5
isoclinism	isoclinic class of #94229
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7236
commutator	C₂×M₁₆	C₄²
Frattini, cosocle	#2931	C₂²
prime radical	#532	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #2931 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

205 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 48 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fgi,d^2=i,e^2=fgi,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,g^a=ghi,h^a=hi,d^b=dg,f^b=fi,g^b=gi,h^b=hi,d^c=dh,f^c=fi,g^c=gi,f^d=fi,g^d=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1428);