#187522

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	#62827	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2036 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:

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

Polycyclic presentation:

2^9|a^2=g,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, 94718);