#135671

order	512
nilpotency class	4
isoclinism	isoclinic class of #134690
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#106
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ei,c^2=d,d^2=g,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=ch,f^b=fi,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42867);