#244627

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	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

932 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^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, 151823);