#478275

order	512
nilpotency class	3
isoclinism	isoclinic class of #476654
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=i,c^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385471);