#8088069

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

4720 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7995265);