#109575

order	512
nilpotency class	3
isoclinism	isoclinic class of #9109
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, prime radical	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1436
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 560 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=h,g^2=i,b^a=bd,c^a=ce,g^a=gh,c^b=ci,d^b=dh,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, 16771);