#492520

order	512
nilpotency class	4
isoclinism	isoclinic class of D₈×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 539 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309422
maximal normal subgroup of	#72309422

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399716);