#494332

order	512
nilpotency class	4
isoclinism	isoclinic class of #12952
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)(3,14,19,46,20,35,16,42)(4,21,24,43,25,58,22,57)(6,28,26,11,31,41,52,44)(8,32,36,17,37,29,34,18)(9,30,39,23,40,59,38,60)(15,47,48,64,51,33,55,61)(45,54,63,50,56,53,62,49);(1,3,7,20)(2,8)(4,15,25,51)(5,16,27,19)(6,29,31,32)(9,33)(10,34)(11,42)(12,36)(13,37)(14,28)(17,52,18,26)(21,56)(22,48,24,55)(23,54,60,53)(30,50,59,49)(35,41)(38,61)(39,64)(40,47)(43,63)(44,46)(45,58)(57,62);(1,4)(2,9)(3,15)(5,22)(6,30)(7,25)(8,33)(10,38)(11,43)(12,39)(13,40)(14,45)(16,48)(17,53)(18,54)(19,55)(20,51)(21,41)(23,52)(24,27)(26,60)(28,58)(29,50)(31,59)(32,49)(34,61)(35,56)(36,64)(37,47)(42,63)(44,57)(46,62);(1,5,7,27)(2,10,13,12)(3,16,20,19)(4,22,25,24)(6,26,31,52)(8,34,37,36)(9,38,40,39)(11,41,44,28)(14,42,35,46)(15,48,51,55)(17,32,18,29)(21,57,58,43)(23,30,60,59)(33,61,47,64)(45,63,56,62)(49,54,50,53)

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=fhi,g^b=gi,h^b=hi,e^c=ehi,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, #72280442, #72281094, …
maximal normal subgroup of	#72280389, #72280442, #72281094, …
maximal quotient of	C₂×#494332, #72310135, #72321909, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401528);