#219613

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	C₂³	#535
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

476 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)(3,12,10,6)(4,15)(5,19,21,14)(8,18)(9,25,26,20)(13,22)(16,27)(17,24)(23,32)(28,29)(30,31)(33,34)(35,43)(36,47)(37,41)(38,52)(39,45)(40,50)(42,51)(44,58)(46,54)(48,62)(49,61)(53,63)(55,59)(56,64)(57,60);(1,3,6,2,7,10,12,11)(4,8,22,13,18,15,17,24)(5,19,20,26,21,14,25,9)(16,28,32,31,30,23,29,27)(33,35)(34,39)(36,48)(37,45)(38,46)(40,55)(41,43)(42,54)(44,49)(47,58)(50,63)(51,60)(52,57)(53,56)(59,64)(61,62);(1,4,12,22,7,18,6,17)(2,8,10,24,11,15,3,13)(5,16,25,32,21,30,20,29)(9,23,19,31,26,28,14,27)(33,36,38,50,37,49,51,64)(34,40,42,47,41,56,52,61)(35,44,46,59,45,48,60,63)(39,53,54,62,43,55,57,58);(1,5,6,20,7,21,12,25)(2,9,10,19,11,26,3,14)(4,16,17,29,18,30,22,32)(8,23,24,31,15,28,13,27)(34,41)(35,45)(40,56)(42,52)(44,48)(46,60)(47,61)(59,63)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=efi,d^2=ehi,e^2=h,f^2=hi,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, 126809);