#100051

order	512
nilpotency class	3
isoclinism	isoclinic class of #100051
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

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	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 400 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7247);