#99932

order	512
nilpotency class	3
isoclinism	isoclinic class of #9279
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#386
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

405 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 112 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=gh,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eh,f^c=fgh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7128);