#9726994

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4175 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9634190);