#147145

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×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	#1923
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ 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:

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

Polycyclic presentation:

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

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, 54341);