#254502

order	512
nilpotency class	3
isoclinism	isoclinic class of #254491
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	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

797 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:

(2,6)(8,15)(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);(1,2)(3,9,12,16)(4,6)(5,8)(7,11,10,14)(13,15)(18,23)(19,27,29,35)(24,37,36,39)(25,30)(26,34)(28,41,42,46)(38,47,43,48)(40,44);(1,3,4,12)(2,7,6,10)(5,14,13,11)(8,16,15,9)(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);(1,2)(3,10)(4,6)(5,8)(7,12)(9,14)(11,16)(13,15)(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|b^2=eh,c^2=i,d^2=hi,g^2=h,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,d^c=di,e^c=eh,g^c=gh,f^d=fh

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, 161698);