#278657

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	#19496	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

543 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)(3,10)(4,15)(5,8)(6,9)(7,12)(11,13)(14,16)(17,18,22,27,23,20,29,26)(19,28,30,24,21,31,32,25);(1,2,5,15,6,9,4,8)(3,11,13,10,14,7,12,16)(17,19,29,32,23,21,22,30)(18,24,27,31,20,25,26,28);(1,2)(3,11)(4,8)(5,15)(6,9)(7,14)(10,13)(12,16)(17,20)(18,23)(19,24)(21,25)(22,27)(26,29)(28,30)(31,32);(1,3,5,13,6,14,4,12)(2,7,15,16,9,11,8,10)(17,21,22,32,23,19,29,30)(18,25,26,31,20,24,27,28)

Polycyclic presentation:

2^9|a^2=f,b^2=eh,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, 185853);