#146960

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

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	#1832
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

512 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)(5,6,10,16,11,17,32,46,12,18,33,47,9,15,30,44)(7,19,23,48,21,36,53,45,22,42,31,59,28,13,34,37)(8,14,26,40,25,39,55,64,27,41,57,66,29,43,58,67)(20,49,52,65,24,35,54,62,51,38,60,63,50,61,56,68);(2,4)(5,7)(6,13)(8,24,27,50)(9,28)(10,31)(11,21)(12,22)(14,38,41,49)(15,42)(16,45)(17,19)(18,36)(20,25,51,29)(23,33)(26,56,57,54)(30,53)(32,34)(35,39,61,43)(37,47)(40,65,66,63)(44,48)(46,59)(52,58,60,55)(62,67,68,64);(5,8)(6,14)(7,20,22,51)(9,29)(10,26)(11,25)(12,27)(13,35,36,61)(15,43)(16,40)(17,39)(18,41)(19,38,42,49)(21,24,28,50)(23,52,31,60)(30,58)(32,55)(33,57)(34,56,53,54)(37,62,45,68)(44,67)(46,64)(47,66)(48,63,59,65)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=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, 54156);