#100529

order	512
nilpotency class	3
isoclinism	isoclinic class of #100235
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	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

450 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=h,d^2=gh,e^2=i,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7725);