#7806757

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

4748 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)(9,10,13,14)(11,17,19,15)(12,16,21,20)(18,23,24,22)(25,26,30,33)(27,34,29,31)(28,32,37,40)(35,39,36,38);(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,27)(26,31)(28,36)(29,30)(32,39)(33,34)(35,37)(38,40);(10,14)(12,21)(15,17)(18,24)(27,29)(31,34)(35,36)(38,39);(9,10)(11,15)(12,16)(13,14)(17,19)(18,22)(20,21)(23,24)(25,26)(27,34)(28,32)(29,31)(30,33)(35,39)(36,38)(37,40);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,28)(26,32)(27,35)(29,36)(30,37)(31,38)(33,40)(34,39);(1,3,4,7)(2,5,6,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24)(25,29,30,27)(26,31,33,34)(28,36,37,35)(32,38,40,39)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7713953);