#278666

order	512
nilpotency class	3
isoclinism	isoclinic class of #278635
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3038	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

526 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185862);