#96991

order	512
nilpotency class	3
isoclinism	isoclinic class of #96990
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	C₂×#127	C₂×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)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24)(25,26,29,32,30,33,28,34)(27,35,37,40,36,31,38,39);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,36)(26,31,33,35)(28,37,29,38)(32,40,34,39);(3,8)(5,7)(11,17)(14,20)(18,23)(21,24)(26,32)(27,36)(28,29)(31,39)(33,34)(35,40)

Polycyclic presentation:

2^9|a^2=f,b^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, 4187);