#289305

order	512
nilpotency class	3
isoclinism	isoclinic class of #289291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#540
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

513 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 196501);