#264008

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	#3171	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

736 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,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, 171204);