#147551

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
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

1002 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)(5,6,10,17,11,15,31,42,12,18,32,45,9,16,30,44)(7,19,22,46,21,13,33,36,23,35,50,60,29,37,56,61)(8,14,27,40,26,39,54,63,28,41,55,64,25,43,53,65)(20,38,48,62,47,34,57,58,49,59,66,68,24,51,52,67);(5,7)(6,13)(8,24)(9,29)(10,22)(11,21)(12,23)(14,38)(15,35)(16,19)(17,36)(18,37)(20,26)(25,49)(27,52)(28,47)(30,56)(31,33)(32,50)(34,39)(40,62)(41,59)(42,60)(43,51)(44,46)(45,61)(48,54)(53,66)(55,57)(58,63)(64,68)(65,67);(2,4)(5,8)(6,14)(7,20)(9,25)(10,27)(11,26)(12,28)(13,34)(15,39)(16,43)(17,40)(18,41)(19,38)(21,47)(22,48)(23,49)(24,29)(30,53)(31,54)(32,55)(33,57)(35,59)(36,58)(37,51)(42,63)(44,65)(45,64)(46,62)(50,66)(52,56)(60,68)(61,67)

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,d^a=di,c^b=cf,d^b=di,f^b=fi,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, 54747);