#7859269

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	#59117	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4440 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7766465);