#7970266

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

4496 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7877462);