#7806730

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

4551 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)(11,14)(12,20)(13,15)(16,18)(17,22)(19,24)(21,23)(25,26)(27,31)(28,37)(29,33)(30,34)(32,39)(35,36)(38,40);(1,3,5,8)(2,6,4,7)(9,11,13,18)(10,14,15,16)(12,21,22,19)(17,24,20,23)(25,27)(26,31)(28,38)(29,39)(30,35)(32,33)(34,36)(37,40);(10,15)(11,18)(17,20)(19,21)(27,35)(29,38)(31,36)(33,40);(1,2,5,4)(3,6,8,7)(9,11)(10,16)(12,19)(13,18)(14,15)(17,24)(20,23)(21,22)(25,28)(26,32)(27,29)(30,39)(31,40)(33,36)(34,37)(35,38);(1,2,5,4)(3,6,8,7)(9,12)(10,17)(11,19)(13,22)(14,23)(15,20)(16,24)(18,21)(25,26)(27,36)(28,32)(29,33)(30,34)(31,35)(37,39)(38,40);(1,4,5,2)(3,7,8,6)(9,10)(11,16)(12,17)(13,15)(14,18)(19,24)(20,22)(21,23)(25,29)(26,33)(27,28)(30,38)(31,37)(32,36)(34,40)(35,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,e^2=g,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, 7713926);