#494321

order	512
nilpotency class	4
isoclinism	isoclinic class of #12948
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1864
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1864
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1037 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280389, #72280652, #72281111, …
maximal normal subgroup of	#72280389, #72280652, #72281111, …
maximal quotient of	C₂×#494321, #72310149, #72321487, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401517);