#6486330

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470185
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1347 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,31,35,32,36,30,34)(27,29,28,33,38,40,37,39);(2,5)(6,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27)(26,33)(28,30)(29,35)(31,37)(32,38)(34,40)(36,39);(11,17)(14,21)(18,19)(22,23)(25,28,32,37)(26,29,36,40)(27,30,38,31)(33,35,39,34);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(25,29,32,40)(26,28,36,37)(27,35,38,34)(30,39,31,33);(1,3,4,7)(2,6,5,8)(10,16)(14,21)(15,24)(22,23)(25,30,32,31)(26,34,36,35)(27,28,38,37)(29,33,40,39)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=hi,e^2=fgi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,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, 6393526);