#96995

order	512
nilpotency class	3
isoclinism	isoclinic class of #96990
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	#2537	C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

572 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 92 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=g,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=eg,f^c=fg

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4191);