#118568

order	512
nilpotency class	3
isoclinism	isoclinic class of #118564
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=ei,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25764);