#300315

order	512
nilpotency class	3
isoclinism	isoclinic class of #300269
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62535	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

999 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=fh,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207511);