#512652

order	512
nilpotency class	5
isoclinism	isoclinic class of #512651
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#124	C₂⁴
prime radical	#3500	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×#41 ◁ #3500 ◁ G
LECS	G ▷ #124 ▷ 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)(3,13,16,40)(4,9)(5,21,23,55)(6,25,27,11)(7,12)(8,15,29,46)(10,24)(14,39,44,61)(17,36,26,41)(18,33)(19,50,22,54)(20,52,49,35)(28,45,59,64)(30,47,31,48)(32,43,60,63)(34,51)(37,56)(38,57)(42,58,53,62);(1,3,7,17)(2,8,12,31)(4,14,18,42)(5,15,23,48)(6,26,27,16)(9,32,33,28)(10,36,38,13)(11,29,25,30)(19,43,22,45)(20,53,49,44)(21,41,55,40)(24,46,57,47)(34,58,37,39)(35,60,52,59)(50,62,54,61)(51,63,56,64);(1,4)(2,9)(3,14)(5,22)(6,20)(7,18)(8,28)(10,37)(11,35)(12,33)(13,39)(15,45)(16,44)(17,42)(19,23)(21,54)(24,56)(25,52)(26,53)(27,49)(29,59)(30,60)(31,32)(34,38)(36,58)(40,61)(41,62)(43,48)(46,64)(47,63)(50,55)(51,57);(4,18)(9,33)(14,42)(19,22)(20,49)(28,32)(34,37)(35,52)(39,58)(43,45)(44,53)(50,54)(51,56)(59,60)(61,62)(63,64)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419848);