#187555

order	512
nilpotency class	3
isoclinism	isoclinic class of #187428
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	#62761	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2002 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=ei,f^b=fi,g^b=ghi,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, 94751);