#265051

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×#3341	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

826 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=eh,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, 172247);