#118931

order	512
nilpotency class	3
isoclinism	isoclinic class of #118926
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	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

481 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)(36,41,52,60)(37,44)(38,42)(39,43)(40,47)(46,57,63,61)(48,55)(49,56)(50,54,62,64)(51,59,53,58);(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)(33,35,39,48)(34,40,43,56)(36,50,52,62)(37,49,38,47)(41,57,60,61)(42,55,44,45)(46,51,63,53)(54,58,64,59);(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)(33,36,37,51,39,52,38,53)(34,41,42,58,43,60,44,59)(35,46,47,62,48,63,49,50)(40,54,45,61,56,64,55,57)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=i,f^2=h,g^2=hi,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=eh,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, 26127);