#253960

order	512
nilpotency class	3
isoclinism	isoclinic class of #253891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	C₂×#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

743 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 161156);