#101703

order	512
nilpotency class	3
isoclinism	isoclinic class of #101687
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	#2011	C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

444 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,5)(4,7)(6,8)(9,10,13,16,14,17,12,22)(11,18,21,23,19,24,20,15)(26,30,32,35)(27,33,34,36);(9,11)(10,15,17,23)(12,20)(13,21)(14,19)(16,18,22,24)(25,26,28,35,29,32,37,30)(27,31,36,39,34,40,33,38);(2,5)(3,8)(6,7)(11,19)(15,23)(18,24)(20,21)(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,d^2=gh,e^2=hi,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=ehi,f^c=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8899);