#512655

order	512
nilpotency class	5
isoclinism	isoclinic class of #512655
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	#60823	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂×#41 ◁ #3500 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

809 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)(3,13,17,40)(4,9)(5,21,23,55)(6,25,27,11)(7,12)(8,16,30,47)(10,24)(14,39,43,61)(15,36,26,41)(18,33)(19,50,22,54)(20,52,49,35)(28,46,59,64)(29,45,31,48)(32,42,60,63)(34,51)(37,56)(38,57)(44,58,53,62);(1,3)(2,8)(4,14)(5,16)(6,26)(7,15)(9,32)(10,36)(11,30)(12,29)(13,38)(17,27)(18,44)(19,42)(20,53)(21,41)(22,46)(23,45)(24,47)(25,31)(28,33)(34,58)(35,60)(37,39)(40,55)(43,49)(48,57)(50,62)(51,63)(52,59)(54,61)(56,64);(1,4)(2,9)(3,14)(5,22)(6,20)(7,18)(8,28)(10,37)(11,35)(12,33)(13,39)(15,44)(16,46)(17,43)(19,23)(21,54)(24,56)(25,52)(26,53)(27,49)(29,32)(30,59)(31,60)(34,38)(36,58)(40,61)(41,62)(42,45)(47,64)(48,63)(50,55)(51,57);(3,15)(4,18)(8,29)(9,33)(13,36)(16,45)(17,26)(19,22)(20,49)(30,31)(34,37)(35,52)(40,41)(47,48)(50,54)(51,56)

Polycyclic presentation:

2^9|a^2=f,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,d^c=di,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, 419851);