#9712807

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

4084 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)(5,8)(6,7)(9,10,16,19)(11,20,13,14)(12,21,22,17)(15,23,24,18)(25,26,30,33)(27,34,37,31)(28,29,32,38)(35,40,39,36);(1,3)(2,4)(5,6)(7,8)(9,11,16,13)(10,14,19,20)(12,18,22,23)(15,21,24,17)(25,27,30,37)(26,31,33,34)(28,35,32,39)(29,36,38,40);(1,2,7,6)(3,5,8,4)(9,12,16,22)(10,17,19,21)(11,18,13,23)(14,15,20,24);(1,4,7,5)(2,8,6,3)(9,13,16,11)(10,14,19,20)(12,18,22,23)(15,17,24,21);(1,5,7,4)(2,8,6,3)(9,14)(10,11)(12,15)(13,19)(16,20)(17,18)(21,23)(22,24)(25,28)(26,29)(27,35)(30,32)(31,36)(33,38)(34,40)(37,39);(1,6,7,2)(3,5,8,4)(9,15)(10,18)(11,21)(12,20)(13,17)(14,22)(16,24)(19,23)(25,29)(26,32)(27,36)(28,33)(30,38)(31,39)(34,35)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=i,f^2=i,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, 9620003);