#495582

order	512
nilpotency class	4
isoclinism	isoclinic class of #480454
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

979 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)(3,6)(4,5)(7,8)(9,10)(11,19,23,33)(12,16)(13,26,20,18)(14,17)(15,22)(21,32,36,40)(24,37,34,31)(25,30)(27,29)(28,35)(38,39);(9,11,20,22,14,23,13,27)(10,15,26,33,17,29,18,19)(12,21,34,35,25,36,24,38)(16,28,37,40,30,39,31,32);(2,5)(6,8)(9,12,14,25)(10,16,17,30)(11,21,23,36)(13,24,20,34)(15,28,29,39)(18,31,26,37)(19,32,33,40)(22,35,27,38);(1,3,4,7)(2,6,5,8)(10,17)(15,29)(16,30)(18,26)(19,33)(28,39)(31,37)(32,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,d^2=f,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=eh,g^b=gh,h^b=hi,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311104, #72311129, #72311177, …
maximal normal subgroup of	#72311104, #72311129, #72311177, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402778);