#99545

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle	C₂³	#386
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

463 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 463 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=g,e^2=gh,f^2=i,b^a=bd,c^a=ce,e^a=eg,c^b=ci,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=egh,f^c=fh

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, 6741);