#479027

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×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

787 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=fi,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, 386223);