#219685

order	512
nilpotency class	3
isoclinism	isoclinic class of #219624
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	#19670	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

479 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)(3,6)(4,15,19,13)(5,20)(7,11)(8,12)(9,18,25,17)(10,21)(14,23)(16,29,32,24)(22,26)(27,30,28,31)(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);(1,3,7,12,8,11,6,2)(4,9,17,15,19,25,18,13)(5,20,22,10,23,14,21,26)(16,29,30,27,32,24,31,28)(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)(2,9,12,25)(3,13,11,15)(5,16)(6,17)(7,18)(8,19)(10,24,26,29)(14,27,20,28)(21,30)(22,31)(23,32)(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);(1,5,6,21,8,23,7,22)(2,10,11,20,12,26,3,14)(4,16,17,30,19,32,18,31)(9,24,15,28,25,29,13,27)(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=dhi,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, 126881);