#7877185

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

4337 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)(11,19)(12,18)(13,16)(14,20)(15,17)(21,22)(23,24)(25,26,30,35)(27,31,36,32)(28,37,38,33)(29,34,39,40);(1,2)(3,7)(4,8)(5,6)(9,11,15,13)(10,16,17,19)(12,21,20,23)(14,22,18,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,35)(37,39)(38,40);(1,3,5,4)(2,7,6,8)(10,17)(12,20)(16,19)(21,23)(25,27,30,36)(26,32,35,31)(28,34,38,40)(29,33,39,37);(9,12,15,20)(10,18,17,14)(11,21,13,23)(16,24,19,22)(25,27)(26,31)(28,34)(29,33)(30,36)(32,35)(37,39)(38,40);(9,13,15,11)(10,19,17,16)(12,23,20,21)(14,24,18,22)(25,28,30,38)(26,33,35,37)(27,34,36,40)(29,32,39,31);(1,4,5,3)(2,8,6,7)(9,14,15,18)(10,20,17,12)(11,22,13,24)(16,23,19,21)(25,29)(26,34)(27,33)(28,31)(30,39)(32,38)(35,40)(36,37)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=hi,e^2=h,f^2=g,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, 7784381);