#265066

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	C₂³	#546
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

766 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,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, 172262);