#187873

order	512
nilpotency class	3
isoclinism	isoclinic class of #187872
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62822	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2213 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,15)(14,16)(17,18)(19,29,32,24)(20,33,35,40)(21,26)(22,28)(23,27)(25,36,41,34)(30,44)(31,42,37,39)(38,45)(43,47)(46,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,16)(10,15)(17,19)(18,24)(20,30)(21,31)(22,37)(23,32)(25,38)(26,39)(27,29)(28,42)(33,44)(34,45)(35,46)(36,47)(40,48)(41,43);(1,3)(2,7)(4,13)(5,12)(6,14)(8,16)(9,11)(10,15)(17,20)(18,25)(19,30)(21,36)(22,34)(23,35)(24,38)(26,33)(27,41)(28,40)(29,43)(31,47)(32,46)(37,45)(39,44)(42,48);(1,4,6,5)(2,8,10,9)(3,12,14,13)(7,11,15,16)(17,21,23,22)(18,26,27,28)(19,31,32,37)(20,34,35,36)(24,39,29,42)(25,40,41,33)(30,45,46,47)(38,48,43,44)

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,f^b=fhi,g^b=ghi,d^c=dh,e^c=ehi,f^c=fh,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95069);