#479733

order	512
nilpotency class	3
isoclinism	isoclinic class of #476659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

763 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)(3,5)(4,6)(7,8)(9,10,14,30,15,17,13,19)(11,20,23,16,24,36,21,32)(12,25,27,34,28,18,29,35)(22,38,26,40,39,33,37,31);(1,3,4,7)(2,5,6,8)(9,11,15,24)(10,16,17,32)(12,26,28,37)(13,23,14,21)(18,33,25,38)(19,36,30,20)(22,29,39,27)(31,35,40,34);(10,17)(11,21)(12,27)(13,14)(16,20)(18,34)(23,24)(25,35)(26,37)(28,29)(32,36)(33,38);(2,6)(5,8)(9,12)(10,18)(11,22)(13,29)(14,27)(15,28)(16,31)(17,25)(19,34)(20,33)(21,37)(23,26)(24,39)(30,35)(32,40)(36,38)

Polycyclic presentation:

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

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, 386929);