#100216

order	512
nilpotency class	3
isoclinism	isoclinic class of #100203
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator, prime radical	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 656 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ci,d^b=dg,e^b=eh,f^b=fg,d^c=dh,f^c=fh

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, 7412);