#147083

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

528 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,d^2=hi,e^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,h^a=hi,f^c=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, 54279);