#319611

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

737 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=egi,c^2=i,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,f^c=fh,g^c=gi

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, 226807);