#219673

order	512
nilpotency class	3
isoclinism	isoclinic class of #219600
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	#532	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

453 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=ef,d^2=e,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=di,f^b=fh,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, 126869);