#277922

order	512
nilpotency class	3
isoclinism	isoclinic class of #277891
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

554 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=e,c^2=i,d^2=ei,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gi,d^c=di,f^c=fh,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, 185118);