#206541

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

534 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 534 conjugacy classes of subgroups — too many to draw. Only its 202 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ch,d^b=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113737);