#108785

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle, prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

734 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 734 conjugacy classes of subgroups — too many to draw. Only its 314 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=eh,d^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,g^a=gh,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15981);