#278000

order	512
nilpotency class	3
isoclinism	isoclinic class of #277971
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	C₂³	#548
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 185196);