#147385

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#124
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1924
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

684 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)(5,6,10,16,11,17,34,49,12,18,35,50,9,15,27,43)(7,19,23,51,21,40,55,47,24,38,32,63,22,13,36,39)(8,25,30,58,29,46,61,48,31,44,33,64,28,14,41,45)(20,52,53,66,26,59,60,68,54,37,57,65,56,42,62,67);(5,7)(6,13)(8,26,31,56)(9,22)(10,32)(11,21)(12,24)(14,42,46,59)(15,38)(16,47)(17,19)(18,40)(20,29,54,28)(23,35)(25,52,44,37)(27,55)(30,62,33,60)(34,36)(39,50)(41,57,61,53)(43,51)(45,68,48,67)(49,63)(58,65,64,66);(1,2,3,4)(5,8,27,61,9,28,35,30,12,31,34,41,11,29,10,33)(6,14,43,58,15,44,50,45,18,46,49,64,17,25,16,48)(7,20,36,60,22,56,32,53,24,54,55,62,21,26,23,57)(13,37,63,67,38,59,47,65,40,52,51,68,19,42,39,66)

Polycyclic presentation:

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

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, 54581);