#6481174

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470003
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=ghi,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=ei,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, 6388370);