#492506

order	512
nilpotency class	4
isoclinism	isoclinic class of #492504
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
prime radical	#19968	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

513 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

59 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, 399702);