#189680

order	512
nilpotency class	3
isoclinism	isoclinic class of #189535
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

755 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=i,e^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,d^c=dh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96876);