#187544

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	#62762	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2028 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)(4,23,26,56,28,38,60,55)(5,18,29,45,30,11,42,44)(7,13,32,47,35,20,49,43)(8,15,33,12,36,22,50,19)(10,37,40,64,17,51,39,34)(24,41,57,48,58,52,63,54)(25,59,27,46,31,62,61,53);(1,3)(2,9)(4,17)(5,18)(6,12)(7,13)(8,22)(10,28)(11,30)(14,33)(15,36)(16,19)(20,35)(21,50)(23,24)(25,39)(26,48)(27,52)(29,43)(31,40)(32,44)(34,46)(37,55)(38,58)(41,61)(42,47)(45,49)(51,56)(53,64)(54,60)(57,59)(62,63);(1,4)(2,10)(3,17)(5,24)(6,31)(7,34)(8,27)(9,28)(11,38)(12,40)(13,46)(14,48)(15,41)(16,25)(18,23)(19,39)(20,53)(21,54)(22,52)(26,33)(29,37)(30,58)(32,57)(35,64)(36,61)(42,51)(43,55)(44,59)(45,62)(47,56)(49,63)(50,60);(1,5,8,7)(2,11,15,20)(3,18,22,13)(4,24,27,34)(6,29,33,32)(9,30,36,35)(10,38,41,53)(12,43,14,44)(16,42,50,49)(17,23,52,46)(19,47,21,45)(25,51,60,63)(26,57,31,37)(28,58,61,64)(39,56,54,62)(40,55,48,59)

Polycyclic presentation:

2^9|a^2=g,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,e^c=ei,f^c=fh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72272561, #72272564, #72272570
maximal normal subgroup of	#72272561, #72272564, #72272570

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 94740);