#7883102

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4411 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=h,e^2=h,f^2=gi,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, 7790298);