#263590

order	512
nilpotency class	3
isoclinism	isoclinic class of #263555
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
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

723 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)(5,9)(6,19)(7,13)(8,17)(10,18)(12,28)(14,23)(16,27)(20,26)(21,25)(22,31)(24,30)(29,32)(33,34,40,44)(35,38,43,39)(36,45,51,60)(37,46,55,58)(41,50,47,49)(42,54,59,53)(48,57,52,56)(61,64,62,63);(1,3,5,13)(2,7,9,11)(4,12,16,29)(6,20,18,14)(8,24,25,22)(10,26,19,23)(15,32,27,28)(17,31,21,30)(33,35,38,44,40,43,39,34)(36,47,49,60,51,41,50,45)(37,46,53,59,55,58,54,42)(48,57,63,62,52,56,64,61);(1,4,5,16)(2,8,9,25)(3,12,13,29)(6,21,18,17)(7,22,11,24)(10,27,19,15)(14,31,20,30)(23,28,26,32)(33,36,40,51)(34,41,44,47)(35,45,43,60)(37,52,55,48)(38,49,39,50)(42,57,59,56)(46,62,58,61)(53,64,54,63);(2,9)(4,16)(7,11)(10,19)(12,29)(17,21)(23,26)(30,31)(33,37,38,53,40,55,39,54)(34,42,35,46,44,59,43,58)(36,48,49,63,51,52,50,64)(41,56,45,61,47,57,60,62)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,g^b=gh,d^c=di,f^c=fi,g^c=gh

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, 170786);