#314627

order	512
nilpotency class	3
isoclinism	isoclinic class of #252235
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2482
prime radical	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

530 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=dh,f^b=fh,e^c=eh,f^c=fhi,g^c=gh,f^d=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 221823);