#119122

order	512
nilpotency class	3
isoclinism	isoclinic class of #119122
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

496 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)(3,12,16,8)(4,17,20,27)(5,11)(6,21)(7,10)(9,19,26,22)(13,25)(14,28,15,24)(18,23)(29,31)(30,32)(33,34)(35,44,47,56)(36,48)(37,42)(38,53)(39,43)(40,46,57,52)(41,50)(45,58,49,55)(51,59)(54,60)(61,62,64,63);(1,3)(2,8)(4,18)(5,15)(6,14)(7,16)(9,25)(10,12)(11,28)(13,19)(17,29)(20,31)(21,24)(22,30)(23,27)(26,32)(33,35)(34,40)(36,49)(37,52)(38,46)(39,47)(41,58)(42,56)(43,57)(44,53)(45,51)(48,64)(50,62)(54,63)(55,60)(59,61);(1,4)(2,9,10,26)(3,13)(5,19)(6,22)(7,20)(8,23,12,29)(11,17,21,27)(14,18)(15,31)(16,30)(24,32,28,25)(33,36,37,50,39,51,38,54)(34,41,42,59,43,60,53,48)(35,45,46,62,47,49,52,63)(40,55,44,61,57,58,56,64)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,e^c=ei,f^c=fhi,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, 26318);