#149185

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

524 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)(5,6,9,15,10,16,25,37,11,14,26,35,27,36,49,58)(7,12,19,30,20,31,43,54,18,29,41,52,42,53,62,66)(8,13,23,32,24,33,48,55,22,34,46,56,47,57,64,67)(17,38,39,59,40,60,61,68,21,28,44,50,45,51,63,65);(1,2,3,4)(5,7)(6,12)(8,21)(9,19)(10,20)(11,18)(13,28)(14,29)(15,30)(16,31)(17,22)(23,44)(24,45)(25,43)(26,41)(27,42)(32,50)(33,51)(34,38)(35,52)(36,53)(37,54)(39,46)(40,47)(48,63)(49,62)(55,65)(56,59)(57,60)(58,66)(61,64)(67,68);(5,8)(6,13)(7,17,18,21)(9,23)(10,24)(11,22)(12,28,29,38)(14,34)(15,32)(16,33)(19,39,41,44)(20,40,42,45)(25,48)(26,46)(27,47)(30,50,52,59)(31,51,53,60)(35,56)(36,57)(37,55)(43,61,62,63)(49,64)(54,65,66,68)(58,67)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=e,g^2=h,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,e^b=ei,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, 56381);