#187573

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	#62838	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:

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

Polycyclic presentation:

2^9|a^2=g,c^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,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, 94769);