#99933

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

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

normal subgroups only

This group has 395 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) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,4,7,5,3,6,8)(10,14,16,19)(11,17,18,20)(25,26,29,33,30,32,28,38)(27,34,37,39,36,40,35,31);(1,3,4,8,5,2,6,7)(9,10,12,19,13,16,21,14)(11,15,20,23,18,24,17,22)(25,27)(26,31,32,39)(28,35)(29,37)(30,36)(33,34,38,40);(9,11,13,18)(10,15,16,24)(12,20,21,17)(14,22,19,23)(26,32)(28,29)(31,39)(35,37)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=h,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, 7129);