#7927659

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

4602 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7834855);