#147366

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	#1923
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:

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

Polycyclic presentation:

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

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, 54562);