#513209

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	D₄×D₁₆	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

384 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fg,g^a=gh,h^a=hi,c^b=ci,d^b=de,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260483
maximal normal subgroup of	#72260483

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420405);