#478406

order	512
nilpotency class	3
isoclinism	isoclinic class of #476293
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

827 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,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, 385602);