#6513015

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500019
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	#62822	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3117 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6420211);