#146414

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

884 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,10,18,12,16,33,45,13,19,11,35,9,17,32,47)(7,20,23,49,25,40,55,61,26,38,24,57,22,14,36,39)(8,27,30,58,29,44,60,43,31,46,34,59,28,15,41,42)(21,50,53,64,52,65,62,68,54,37,56,63,51,48,66,67);(2,4)(5,7,11,24,12,25,32,36,13,26,10,23,9,22,33,55)(6,14,18,39,16,20,45,49,19,40,35,61,17,38,47,57)(8,21,30,53,29,52,60,62,31,54,34,56,28,51,41,66)(15,37,43,64,27,48,59,68,44,50,42,63,46,65,58,67);(5,8)(6,15)(7,21)(9,28)(10,34)(11,30)(12,29)(13,31)(14,37)(16,27)(17,46)(18,43)(19,44)(20,48)(22,51)(23,56)(24,53)(25,52)(26,54)(32,60)(33,41)(35,42)(36,62)(38,65)(39,64)(40,50)(45,59)(47,58)(49,68)(55,66)(57,67)(61,63)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,g^a=gi,e^c=ei,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53610);