#219668

order	512
nilpotency class	3
isoclinism	isoclinic class of #219590
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=ef,d^2=ei,e^2=h,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=dh,f^b=fh,d^c=di,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, 126864);