#146839

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1746
prime radical	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

496 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)(5,6,10,18,12,16,33,46,13,19,11,35,9,17,32,47)(7,20,23,49,25,40,54,61,26,38,24,56,22,14,36,39)(8,27,29,58,30,15,41,43,31,42,34,59,28,45,60,44)(21,37,51,63,52,48,62,66,53,65,55,64,50,57,68,67);(5,7,11,24,12,25,32,36,13,26,10,23,9,22,33,54)(6,14,18,39,16,20,46,49,19,40,35,61,17,38,47,56)(8,21,29,51,30,52,41,62,31,53,34,55,28,50,60,68)(15,37,44,64,42,48,58,67,45,65,43,63,27,57,59,66);(2,4)(5,8,9,28,13,31,12,30)(6,15,16,42,19,45,17,27)(7,21,22,50,26,53,25,52)(10,34,32,41,11,29,33,60)(14,37,20,48,40,65,38,57)(18,44,46,58,35,43,47,59)(23,55,36,62,24,51,54,68)(39,64,49,67,61,63,56,66)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,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,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, 54035);