#7859295

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	#39372	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4423 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)(11,19,20,16)(13,21,23,17)(14,18,24,22)(26,28)(31,33)(32,36)(38,39);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,19)(13,18)(14,21)(15,20)(17,24)(22,23);(1,3,5,4)(2,7,6,8)(10,12)(11,20)(17,21)(18,22)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(9,12,15,10)(11,19,20,16)(13,21,23,17)(14,22,24,18)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,3,5,4)(2,7,6,8)(9,13)(10,17)(11,18)(12,21)(14,19)(15,23)(16,24)(20,22)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(1,4,5,3)(2,8,6,7)(9,14)(10,18)(11,21)(12,22)(13,16)(15,24)(17,20)(19,23)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=h,e^2=gh,f^2=gh,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, 7766491);