#152804

order	512
nilpotency class	5
isoclinism	isoclinic class of #151115
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2177
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

174 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 174 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=dh,d^2=i,e^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^b=ei,d^c=di,e^c=egi,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60000);