#492576

order	512
nilpotency class	4
isoclinism	isoclinic class of D₈×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	D₄×D₁₆	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

489 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 489 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Polycyclic presentation:

2^9|b^2=gi,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,h^a=hi,d^b=de,e^b=eg,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309684
maximal normal subgroup of	#72309684

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399772);