#152661

order	512
nilpotency class	5
isoclinism	isoclinic class of #149897
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×AΣL(1,𝔽₁₆)
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂×#127	C₂³
prime radical	#23908	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

652 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 652 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 20
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 20 points:

(1,2)(3,4)(5,6,9,13)(7,12,17,20)(8,18,10,14)(11,15,19,16);(6,11,19,13,14,20,12,18)(7,15)(8,16)(9,17);(2,4)(5,7,10,15)(6,12,14,19)(8,16,9,17)(11,13,20,18)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ci,d^b=dgi,f^b=fgi,g^b=gh,f^d=fhi,g^d=gi,g^f=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59857);