#152833

order	512
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

226 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 86 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)(5,6,8,7)(10,11)(12,13)(14,25)(15,30)(17,27)(18,22)(19,35)(20,34)(21,24)(23,36)(26,33)(28,29)(31,39)(32,38)(37,40);(5,7,8,6)(9,10,17,32,14,19,26,23,30,36,29,35,39,38,12,11,16,21,13,18,31,37,28,34,15,20,33,40,25,22,27,24);(2,4)(9,11,12,22,25,35,29,20,15,23,26,37,31,32,17,21,16,24,27,38,39,40,33,36,30,34,28,19,14,18,13,10)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,c^2=d,d^2=gh,e^2=g,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,h^a=hi,c^b=ci

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, 60029);