#265215

order	512
nilpotency class	3
isoclinism	isoclinic class of #265163
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

809 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

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=ghi,c^b=cg,d^b=dh,f^b=fhi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

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, 172411);