#265055

order	512
nilpotency class	3
isoclinism	isoclinic class of #265019
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×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

833 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:

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

Polycyclic presentation:

2^9|b^2=ei,c^2=i,d^2=e,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,f^b=fhi,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

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, 172251);