#244649

order	512
nilpotency class	3
isoclinism	isoclinic class of #244583
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	#3506	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

941 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)(3,11)(4,15)(5,18)(6,10)(7,13)(8,16)(9,21)(12,23)(14,25)(17,26)(19,22)(20,24)(27,28)(29,31)(30,32)(35,44)(37,38)(40,45)(42,52)(46,61)(47,48)(49,51)(53,63)(54,58)(55,56)(59,62)(60,64);(1,3)(2,7)(4,12)(5,19)(6,14)(8,23)(9,25)(10,22)(11,18)(13,21)(15,27)(16,30)(17,29)(20,28)(24,31)(26,32)(33,34,39,43)(35,45,47,58)(36,41,50,57)(37,42,38,52)(40,44,54,48)(46,60,62,63)(49,55,51,56)(53,59,64,61);(7,22)(11,25)(23,31)(27,32)(33,35,39,47)(34,40,43,54)(36,46,50,62)(37,48,38,44)(41,53,57,64)(42,58,52,45)(49,59,51,61)(55,60,56,63);(1,4,6,17)(2,8,10,24)(3,12,14,29)(5,20,21,16)(7,23,22,31)(9,26,18,15)(11,27,25,32)(13,30,19,28)(33,36)(34,41)(35,46)(37,51)(38,49)(39,50)(40,53)(42,56)(43,57)(44,59)(45,60)(47,62)(48,61)(52,55)(54,64)(58,63)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151845);