#119103

order	512
nilpotency class	3
isoclinism	isoclinic class of #119098
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

493 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 26299);