#102206

order	512
nilpotency class	3
isoclinism	isoclinic class of #102154
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	#390
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

346 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=eg,d^2=gh,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cgi,d^b=dh,f^b=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9402);