#118999

order	512
nilpotency class	3
isoclinism	isoclinic class of #118997
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

494 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)(35,43,46,56)(36,40)(37,41)(38,42)(39,45,53,50)(44,59,47,58)(48,54)(49,55)(51,57)(52,61,63,60)(62,64);(1,3)(2,8)(4,18)(5,16)(6,15)(7,14)(9,25)(10,12)(11,28)(13,20)(17,32)(19,31)(21,24)(22,30)(23,26)(27,29)(33,35)(34,39)(36,47)(37,50)(38,46)(40,52)(41,56)(42,53)(43,57)(44,49)(45,51)(48,60)(54,59)(55,63)(58,64)(61,62);(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)(33,36,37,48,38,49,51,62)(34,40,41,54,42,55,57,64)(35,44,45,60,46,47,50,61)(39,52,43,58,53,63,56,59)

Polycyclic presentation:

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

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, 26195);