#368889

order	512
nilpotency class	3
isoclinism	isoclinic class of #368888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14023
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1646 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)(3,15)(4,20,25,10)(5,26)(6,14)(7,12)(8,13)(9,18)(11,27)(16,34,43,47)(17,32)(19,39)(21,31,50,57)(22,51,53,55)(23,36,24,37)(28,40)(29,33)(30,38)(35,52,48,54)(41,44)(42,56,62,64)(45,59)(46,58)(49,63,61,60);(1,3)(2,9)(4,21,25,50)(5,17)(6,29)(7,18)(8,19)(10,34,20,47)(11,32)(12,39)(13,33)(14,15)(16,24,43,23)(22,49,53,61)(26,41)(27,46)(28,45)(30,44)(31,37,57,36)(35,60,48,63)(38,59)(40,58)(42,52,62,54)(51,56,55,64);(1,4,7,24,8,25,6,23)(2,10,12,36,13,20,14,37)(3,16,29,50,19,43,18,21)(5,22,27,52,28,53,30,54)(9,31,15,34,33,57,39,47)(11,35,40,51,38,48,26,55)(17,42,44,61,45,62,46,49)(32,56,41,60,59,64,58,63);(1,5,8,28)(2,11,13,38)(3,17,19,45)(4,22,25,53)(6,30,7,27)(9,32,33,59)(10,35,20,48)(12,40,14,26)(15,41,39,58)(16,42,43,62)(18,46,29,44)(21,49,50,61)(23,54,24,52)(31,56,57,64)(34,60,47,63)(36,51,37,55)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=fg,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^c=hi,e^d=ei

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276085);