#187880

order	512
nilpotency class	3
isoclinism	isoclinic class of #187878
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	#62823	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2060 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=i,f^2=h,b^a=be,c^a=cf,d^a=dh,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95076);