#102364

order	512
nilpotency class	3
isoclinism	isoclinic class of #102147
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

411 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 411 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9560);