#187565

order	512
nilpotency class	3
isoclinism	isoclinic class of #187442
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	#62830	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2181 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)(3,12,15,7)(4,9)(5,11)(6,10)(8,14,16,13)(17,18,22,27,23,26,21,28)(19,29,31,24,32,40,37,39)(20,33,36,25,34,41,35,42)(30,44,46,48,45,38,47,43);(1,3)(2,7)(4,13)(5,14)(6,15)(8,11)(9,16)(10,12)(17,19)(18,24)(20,30)(21,37)(22,31)(23,32)(25,38)(26,39)(27,40)(28,29)(33,48)(34,45)(35,47)(36,46)(41,43)(42,44);(1,3)(2,8)(4,14)(5,13)(6,15)(7,11)(9,12)(10,16)(17,20)(18,25)(19,30)(21,36)(22,35)(23,34)(24,38)(26,42)(27,33)(28,41)(29,43)(31,47)(32,45)(37,46)(39,44)(40,48);(1,4,6,5)(2,9,10,11)(3,13,15,14)(7,16,12,8)(18,26)(20,34)(24,39)(27,28)(29,40)(30,45)(35,36)(46,47)

Polycyclic presentation:

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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94761);