#264018

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	#19019	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

774 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)(3,7)(4,8)(5,9)(6,10)(11,15)(12,14)(13,16)(17,18,24,30)(19,31,35,47)(20,36,33,48)(21,39,41,46)(22,29,23,28)(25,37,44,34)(26,42,45,38)(27,32,43,40);(1,3)(2,7)(4,12)(5,11)(6,13)(8,14)(9,15)(10,16)(17,19,22,34,24,35,23,37)(18,25,28,31,30,44,29,47)(20,32,38,21,33,40,42,41)(26,27,36,39,45,43,48,46);(2,8)(3,11)(4,5)(9,10)(12,13)(14,15)(17,20)(18,26)(19,32)(21,34)(22,38)(23,42)(24,33)(25,43)(27,44)(28,36)(29,48)(30,45)(31,46)(35,40)(37,41)(39,47);(1,4,6,5)(2,8,10,9)(3,12,13,11)(7,14,16,15)(17,21,22,40,24,41,23,32)(18,27,28,39,30,43,29,46)(19,33,34,42,35,20,37,38)(25,36,31,45,44,48,47,26)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=ehi,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, 171214);