#187524

order	512
nilpotency class	3
isoclinism	isoclinic class of #187410
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	#62835	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2160 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:

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

Polycyclic presentation:

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

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, 94720);