#100053

order	512
nilpotency class	3
isoclinism	isoclinic class of #100051
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	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 416 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)(11,17)(12,20,22,15)(13,19)(14,16)(18,21,24,23)(25,26,29,32)(27,30)(28,33,37,38)(31,36)(34,40)(35,39);(1,3,5,8,4,2,6,7)(9,11,17,10,14,19,13,16)(12,18,21,15,22,24,23,20)(25,27)(26,30,32,40)(28,36)(29,34)(31,38,39,33)(35,37);(2,3)(7,8)(9,12)(10,15)(11,18)(13,23)(14,22)(16,20)(17,21)(19,24)(25,27)(26,31)(28,36)(29,34)(30,38)(32,39)(33,40)(35,37)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,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, 7249);