#99082

order	512
nilpotency class	3
isoclinism	isoclinic class of #98936
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#380
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

342 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=i,d^2=g,e^2=i,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,e^c=ei,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278);