#6490447

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

1390 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)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26)(27,38)(28,40)(29,34)(30,36)(31,35)(32,37)(33,39);(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27)(26,33)(28,31)(29,32)(30,39)(34,36)(35,38)(37,40);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(25,28,32,39)(26,33,37,40)(27,30,29,31)(34,36,38,35);(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,29,32,27)(26,34,37,38)(28,30,39,31)(33,35,40,36);(1,3,4,7)(2,6,5,8)(10,18)(15,24)(16,17)(22,23)(25,30,32,31)(26,35,37,36)(27,39,29,28)(33,38,40,34)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,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, 6397643);