#250494

order	512
nilpotency class	3
isoclinism	isoclinic class of #250455
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61940	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

926 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157690);