#146926

order	512
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#427
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1807
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

484 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=h,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gh,h^a=hi,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54122);