#147536

order	512
nilpotency class	3
isoclinism	isoclinic class of #2010
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1955
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

1006 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)(5,6,10,17,11,16,32,45,12,18,33,47,9,15,30,44)(7,19,23,48,21,36,53,46,22,37,31,60,29,13,34,38)(8,14,27,41,26,42,57,66,28,43,59,67,25,40,56,64)(20,49,52,65,50,35,58,62,51,39,61,63,24,54,55,68);(2,4)(5,7)(6,13)(8,24)(9,29)(10,31)(11,21)(12,22)(14,39)(15,37)(16,19)(17,46)(18,36)(20,26)(23,33)(25,51)(27,58)(28,50)(30,53)(32,34)(35,40)(38,47)(41,65)(42,54)(43,49)(44,48)(45,60)(52,56)(55,59)(57,61)(62,66)(63,67)(64,68);(5,8)(6,14)(7,20)(9,25)(10,27)(11,26)(12,28)(13,35)(15,40)(16,42)(17,41)(18,43)(19,39)(21,50)(22,51)(23,52)(24,29)(30,56)(31,61)(32,57)(33,59)(34,55)(36,54)(37,49)(38,62)(44,64)(45,66)(46,68)(47,67)(48,63)(53,58)(60,65)

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,f^a=fi,c^b=cf,d^b=di,e^b=ei,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, 54732);