#7988206

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

4511 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)(11,14)(12,17)(13,16)(15,19)(18,22)(20,24)(21,23)(25,26)(27,35)(28,34)(29,32)(30,36)(31,33)(37,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,10,14,11)(12,15,19,17)(13,18,22,16)(20,24,21,23)(25,27,31,29)(26,32,33,35)(28,37,36,39)(30,38,34,40);(10,11)(12,19)(16,18)(20,21)(26,33)(28,36)(32,35)(37,39);(9,11,14,10)(12,17,19,15)(13,18,22,16)(20,24,21,23)(25,28,31,36)(26,34,33,30)(27,37,29,39)(32,40,35,38);(9,12)(10,15)(11,17)(13,21)(14,19)(16,24)(18,23)(20,22)(25,29,31,27)(26,35,33,32)(28,39,36,37)(30,40,34,38);(1,3,4,7)(2,5,6,8)(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,30,31,34)(26,36,33,28)(27,38,29,40)(32,39,35,37)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=h,e^2=hi,f^2=gi,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, 7895402);