#264014

order	512
nilpotency class	3
isoclinism	isoclinic class of #264003
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	#121	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

719 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:

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

Polycyclic presentation:

2^9|a^2=hi,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=gh,c^b=cg,f^b=fh,g^b=ghi,d^c=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 171210);