#6434183

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2056 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341379);