#147371

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

710 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=ei,d^b=di,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54567);