#101802

order	512
nilpotency class	3
isoclinism	isoclinic class of #101799
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

394 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=gi,d^2=gh,e^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=egh,f^b=fh,d^c=dgh,e^c=eghi,f^c=fgh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8998);