#187561

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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1980 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,14,9,3,16,21)(4,23,27,56,29,43,60,55)(5,18,30,46,31,11,42,45)(7,13,33,48,35,20,50,44)(8,15,34,12,36,22,51,19)(10,37,40,58,17,52,39,24)(25,59,28,47,32,64,63,38)(26,41,61,49,62,53,57,54);(1,3)(2,9)(4,17)(5,18)(6,12)(7,13)(8,22)(10,29)(11,31)(14,34)(15,36)(16,19)(20,35)(21,51)(23,26)(24,47)(25,39)(27,49)(28,53)(30,44)(32,40)(33,45)(37,55)(38,58)(41,63)(42,48)(43,62)(46,50)(52,56)(54,60)(57,64)(59,61);(1,4,8,28)(2,10,15,41)(3,17,22,53)(5,26,7,24)(6,32,34,27)(9,29,36,63)(11,43,20,38)(12,40,14,49)(13,47,18,23)(16,25,51,60)(19,39,21,54)(30,37,33,61)(31,62,35,58)(42,52,50,57)(44,55,45,59)(46,64,48,56);(1,5,8,7)(2,11,15,20)(3,18,22,13)(4,24,28,26)(6,30,34,33)(9,31,36,35)(10,38,41,43)(12,44,14,45)(16,42,51,50)(17,47,53,23)(19,48,21,46)(25,57,60,52)(27,37,32,61)(29,58,63,62)(39,64,54,56)(40,59,49,55)

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,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, 94757);