#512668

order	512
nilpotency class	5
isoclinism	isoclinic class of #512666
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	#127	C₂⁴
prime radical	#60823	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ #3500 ◁ G
LECS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

812 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)(3,14,17,8)(4,9)(5,10)(6,12,27,11)(7,13)(15,40,43,28)(16,41,45,29)(18,30,26,31)(19,47,48,55)(20,32)(21,34)(22,33,50,39)(23,35)(24,37,53,36)(25,38)(42,61,63,54)(44,62,64,56)(46,57,52,58)(49,59,51,60);(1,3)(2,8)(4,19)(5,16)(6,26)(7,18)(9,28)(10,29)(11,30)(12,14)(13,31)(15,21)(17,27)(20,44)(22,43)(23,42)(24,52)(25,46)(32,54)(33,40)(34,55)(35,56)(36,57)(37,41)(38,58)(39,47)(45,53)(48,50)(49,63)(51,64)(59,61)(60,62);(1,4)(2,9)(3,15)(5,23)(6,22)(7,21)(8,28)(10,35)(11,39)(12,33)(13,34)(14,40)(16,44)(17,43)(18,19)(20,25)(24,51)(26,48)(27,50)(29,56)(30,47)(31,55)(32,38)(36,59)(37,60)(41,62)(42,46)(45,64)(49,53)(52,63)(54,58)(57,61);(1,5,7,25)(2,10,13,38)(3,16,18,46)(4,20,21,23)(6,24,27,53)(8,29,31,58)(9,32,34,35)(11,36,12,37)(14,41,30,57)(15,42,19,44)(17,45,26,52)(22,49,50,51)(28,54,55,56)(33,59,39,60)(40,61,47,62)(43,63,48,64)

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=hi,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,d^c=di,e^c=ei,f^e=fhi,g^f=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317784, #72318931, #72318941
maximal normal subgroup of	#72317784, #72318931, #72318941

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419864);