#187655

order	512
nilpotency class	3
isoclinism	isoclinic class of #19086
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 52 points:

(1,2)(3,4)(5,6,9,13,10,12,8,14)(7,15,16,11,18,20,17,19)(21,22,26,32,27,30,25,31)(23,33,36,43,37,28,35,44)(24,38,41,29,39,46,40,45)(34,48,51,47,49,52,50,42);(5,7,10,18)(6,11,12,19)(8,17,9,16)(13,20,14,15)(21,23,27,37)(22,28,30,33)(24,34,39,49)(25,36,26,35)(29,42,45,47)(31,44,32,43)(38,48,46,52)(40,51,41,50);(2,4)(5,6,10,12)(7,11,18,19)(8,13,9,14)(15,17,20,16)(21,24)(22,29)(23,34)(25,41)(26,40)(27,39)(28,42)(30,45)(31,46)(32,38)(33,47)(35,51)(36,50)(37,49)(43,48)(44,52);(6,12)(11,19)(13,14)(15,20)(24,39)(29,45)(34,49)(38,46)(40,41)(42,47)(48,52)(50,51)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fhi,g^b=ghi,d^c=dh,e^c=ehi,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, 94851);