#478507

order	512
nilpotency class	3
isoclinism	isoclinic class of #476297
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

807 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)(10,16)(12,26)(13,14)(17,21)(18,31)(19,20)(22,38)(23,24)(27,36)(32,37)(33,34)(35,39);(1,3,4,7)(2,5,6,8)(9,10,13,16,15,20,14,19)(11,21,23,17,25,33,24,34)(12,27,30,36,26,31,29,18)(22,39,40,35,38,32,28,37);(2,6)(5,8)(9,11,15,25)(10,17,20,34)(12,28,26,40)(13,24,14,23)(16,21,19,33)(18,35,36,37)(22,30,38,29)(27,39,31,32);(3,7)(5,8)(9,12,13,29,15,26,14,30)(10,18,16,31,20,36,19,27)(11,22,23,28,25,38,24,40)(17,32,33,35,34,39,21,37)

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,c^2=i,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=def,h^b=hi,d^c=de,e^c=ei,g^c=gi,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, 385703);