#146828

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1746
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

454 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)(5,6,9,14,10,16,26,38,11,15,27,37,28,39,53,63)(7,17,20,41,21,42,46,65,19,12,29,31,45,32,55,58)(8,13,23,34,25,35,51,61,24,36,50,62,52,56,67,68)(18,40,44,64,22,47,49,66,43,30,54,57,48,33,59,60);(5,7)(6,12,15,17)(8,22,24,48)(9,20)(10,21)(11,19)(13,33)(14,31,37,41)(16,32,39,42)(18,25,43,52)(23,49,50,59)(26,46)(27,29)(28,45)(30,56)(34,60)(35,40)(36,47)(38,58,63,65)(44,51,54,67)(53,55)(57,68)(61,64)(62,66);(2,4)(5,8)(6,13)(7,18,19,43)(9,23)(10,25)(11,24)(12,30,17,40)(14,34)(15,36)(16,35)(20,44,29,54)(21,22,45,48)(26,51)(27,50)(28,52)(31,57,41,64)(32,33,42,47)(37,62)(38,61)(39,56)(46,49,55,59)(53,67)(58,60,65,66)(63,68)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,f^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,c^b=ch,d^b=di,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, 54024);