#264990

order	512
nilpotency class	3
isoclinism	isoclinic class of #264939
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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

821 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)(3,10)(4,8)(5,7)(6,12)(9,14)(11,16)(13,15)(17,18,24,28)(19,29,32,43)(20,26,36,42)(21,22,27,23)(25,37,40,31)(30,41,47,45)(33,39,46,48)(34,35,44,38);(1,3,4,12)(2,6,8,10)(5,11,14,13)(7,15,9,16)(17,19,23,37,24,32,22,31)(18,25,27,43,28,40,21,29)(20,33,38,47,36,46,35,30)(26,41,44,48,42,45,34,39);(2,7)(3,11)(5,14)(8,9)(12,13)(15,16)(17,20)(18,26)(19,30)(21,34)(22,35)(23,38)(24,36)(25,39)(27,44)(28,42)(29,45)(31,46)(32,47)(33,37)(40,48)(41,43);(2,8)(6,10)(7,9)(15,16)(17,21,22,28,24,27,23,18)(19,29,31,40,32,43,37,25)(20,34,35,42,36,44,38,26)(30,45,46,48,47,41,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=ei,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,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172186);