#512674

order	512
nilpotency class	5
isoclinism	isoclinic class of #13609
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
socle	C₂	C₂×#2127
prime radical	#60797	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3449 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

802 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=ehi,f^b=fgh,g^b=gi,e^c=ei,f^e=fhi,g^f=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419870);