#148923

order	512
nilpotency class	3
isoclinism	isoclinic class of #2021
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator, prime radical	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1850
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

464 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=di,f^b=fi,g^b=gi,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, 56119);