#112170

order	512
nilpotency class	3
isoclinism	isoclinic class of #112162
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
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

736 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=di,f^c=fh,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19366);