#109493

order	512
nilpotency class	3
isoclinism	isoclinic class of #9103
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

568 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16689);