#7719087

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#44078	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7626283);