#147550

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂×#575	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1000 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)(7,19,22,49,21,37,53,45,23,43,31,59,29,13,34,36)(8,14,27,41,26,40,56,65,28,42,58,67,25,39,55,64)(20,48,51,66,50,62,57,68,52,38,60,63,24,35,54,61);(5,7)(6,13)(8,24)(9,29)(10,31)(11,21)(12,23)(14,38)(15,43)(16,45)(17,19)(18,37)(20,26)(22,33)(25,52)(27,57)(28,50)(30,53)(32,34)(35,40)(36,47)(39,62)(41,66)(42,48)(44,49)(46,59)(51,55)(54,58)(56,60)(61,64)(63,67)(65,68);(2,4)(5,8)(6,14)(7,20)(9,25)(10,27)(11,26)(12,28)(13,35)(15,39)(16,41)(17,40)(18,42)(19,48)(21,50)(22,51)(23,52)(24,29)(30,55)(31,60)(32,56)(33,58)(34,54)(36,61)(37,62)(38,43)(44,64)(45,68)(46,65)(47,67)(49,66)(53,57)(59,63)

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=di,f^b=fi,g^b=gi,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, 54746);