#7879851

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	#36149	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4494 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)(9,10,15,12)(11,13,18,16)(14,17,23,20)(19,21,24,22)(25,26,30,27)(28,31,36,33)(29,32,38,34)(35,39,37,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,18)(10,16,12,13)(14,19,23,24)(17,22,20,21)(26,27)(31,33)(32,34)(39,40);(1,3,5,4)(2,7,6,8)(10,12)(11,18)(17,20)(19,24)(25,27,30,26)(28,33,36,31)(29,34,38,32)(35,40,37,39);(9,12,15,10)(11,13,18,16)(14,20,23,17)(19,21,24,22);(1,4,5,3)(2,8,6,7)(9,13)(10,11)(12,18)(14,22)(15,16)(17,24)(19,20)(21,23)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(9,14,15,23)(10,17,12,20)(11,19,18,24)(13,21,16,22)(25,29,30,38)(26,32,27,34)(28,35,36,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=h,e^2=gh,f^2=i,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, 7787047);