#510873

order	512
nilpotency class	4
isoclinism	isoclinic class of #510872
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31996
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2932 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

441 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)(4,22)(5,12)(6,11)(7,30)(8,14)(9,18)(10,24)(13,33)(16,46)(17,21)(19,36)(20,51)(23,37)(25,60)(26,39)(27,41)(28,40)(29,42)(31,35)(32,38)(34,50)(43,55)(44,47)(45,49)(48,53)(52,63)(54,62)(56,57)(58,64)(59,61);(1,3,8,21)(2,9,14,36)(4,23,26,58)(5,19,6,18)(7,31,33,52)(10,37,40,64)(11,17,12,15)(13,43,30,51)(16,47,50,25)(20,27,55,29)(22,56,39,62)(24,54,28,57)(32,49,59,53)(34,61,46,38)(35,42,63,41)(44,48,60,45);(1,4,6,28,8,26,5,24)(2,10,11,22,14,40,12,39)(3,16,19,53,21,50,18,49)(7,32,29,47,33,59,27,25)(9,34,15,45,36,46,17,48)(13,44,41,61,30,60,42,38)(20,54,31,58,55,57,52,23)(35,62,43,64,63,56,51,37);(1,2)(3,17)(4,10)(5,12)(6,11)(7,13)(8,14)(9,19)(15,21)(16,48)(18,36)(20,51)(22,28)(23,56)(24,39)(25,38)(26,40)(27,42)(29,41)(30,33)(31,35)(32,44)(34,53)(37,54)(43,55)(45,50)(46,49)(47,61)(52,63)(57,64)(58,62)(59,60)

Polycyclic presentation:

2^9|b^2=i,c^2=efh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gh,h^b=hi,f^c=fi,g^c=ghi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320498, #72320520, #72320582
maximal normal subgroup of	#72320498, #72320520, #72320582
maximal quotient of	#72321300, #72321483, #72321693, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418069);