#263898

order	512
nilpotency class	3
isoclinism	isoclinic class of #263859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3511	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

769 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,8)(4,14)(5,10)(6,11)(7,12)(9,15)(13,16)(17,18,23,28)(19,25,31,40)(20,32,36,47)(21,37,22,27)(24,38,41,30)(26,35,44,34)(29,45,33,39)(42,48,43,46);(1,3)(2,8)(4,12)(5,13)(6,9)(7,14)(10,16)(11,15)(17,19,22,38,23,31,21,30)(18,24,37,40,28,41,27,25)(20,29,35,48,36,33,34,46)(26,39,47,43,44,45,32,42);(1,4,7,8)(2,9,12,13)(3,11,14,10)(5,15,6,16)(17,19)(18,25)(20,33)(21,30)(22,38)(23,31)(24,37)(26,43)(27,41)(28,40)(29,36)(32,39)(34,48)(35,46)(42,44)(45,47);(1,5,7,6)(2,10,12,11)(3,13,14,9)(4,15,8,16)(17,20,22,35,23,36,21,34)(18,26,37,47,28,44,27,32)(19,29,38,48,31,33,30,46)(24,39,40,43,41,45,25,42)

Polycyclic presentation:

2^9|a^2=h,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,f^a=fi,c^b=cg,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, 171094);