#494477

order	512
nilpotency class	4
isoclinism	isoclinic class of #13005
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1874
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1874
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1367 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=h,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=eh,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280321, #72280389, #72281034, …
maximal normal subgroup of	#72280321, #72280389, #72281034, …
maximal quotient of	C₂×#494477, #72310113, #72342525

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401673);