#512650

order	512
nilpotency class	5
isoclinism	isoclinic class of #512647
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	#3449	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #3449 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

803 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)(3,15,19,32)(4,10)(5,11)(6,27)(7,13,30,40)(8,14)(9,18,33,50)(12,29,42,26)(16,44,47,57)(17,43,49,34)(20,39,21,45)(22,52,48,60)(23,35)(24,37,54,62)(25,38)(28,55)(31,51,58,64)(36,53,41,56)(46,61,63,59);(1,3,8,21)(2,9,14,34)(4,22,25,16)(5,18,6,17)(7,19,30,20)(10,31,38,59)(11,39,27,32)(12,15,42,45)(13,43,40,33)(23,46,28,51)(24,48,54,47)(26,50,29,49)(35,60,55,57)(36,52,41,44)(37,61,62,58)(53,63,56,64);(1,4)(2,10)(3,16)(5,23)(6,28)(7,24)(8,25)(9,31)(11,35)(12,41)(13,37)(14,38)(15,44)(17,46)(18,51)(19,47)(20,48)(21,22)(26,53)(27,55)(29,56)(30,54)(32,57)(33,58)(34,59)(36,42)(39,60)(40,62)(43,61)(45,52)(49,63)(50,64);(1,5,8,6)(2,11,14,27)(3,17,21,18)(4,23,25,28)(7,26,30,29)(9,32,34,39)(10,35,38,55)(12,40,42,13)(15,43,45,33)(16,46,22,51)(19,49,20,50)(24,53,54,56)(31,57,59,60)(36,37,41,62)(44,61,52,58)(47,63,48,64)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284730, #72287238
maximal normal subgroup of	#72284730, #72287238

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419846);