#146840

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,f^b=fi,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, 54036);