#264813

order	512
nilpotency class	3
isoclinism	isoclinic class of #264779
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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

780 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)(3,11)(4,8)(5,9)(6,10)(7,14)(12,15)(13,16)(17,18)(19,25)(20,33)(21,36)(22,29)(23,28)(24,31)(26,37)(27,34)(30,44)(32,40)(35,39)(38,43)(41,45)(42,48)(46,47);(1,2)(3,12)(4,8)(5,9)(6,10)(7,13)(11,15)(14,16)(17,19,22,31,23,20,34,32)(18,24,27,25,28,40,29,33)(21,30,37,45,38,35,48,46)(26,39,36,41,42,44,43,47);(1,3,6,15)(2,7,10,16)(4,13,5,14)(8,11,9,12)(17,20)(18,25)(19,23)(21,35)(22,32)(24,29)(26,41)(27,40)(28,33)(30,38)(31,34)(36,44)(37,46)(39,43)(42,47)(45,48);(1,4,6,5)(2,8,10,9)(3,13,15,14)(7,11,16,12)(17,21,34,48,23,38,22,37)(18,26,29,43,28,42,27,36)(19,30,32,46,20,35,31,45)(24,39,33,47,40,44,25,41)

Polycyclic presentation:

2^9|b^2=e,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,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,f^c=fi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172009);