#7806841

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

4768 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)(11,15)(12,21)(13,17)(14,18)(16,23)(19,20)(22,24)(25,26,29,30)(27,33,34,31)(28,32,36,38)(35,40,37,39);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,22)(13,23)(14,19)(16,17)(18,20)(21,24)(26,30)(27,34)(28,36)(39,40);(11,19)(13,22)(15,20)(17,24)(25,27,29,34)(26,31,30,33)(28,37,36,35)(32,40,38,39);(9,12)(10,16)(11,13)(14,23)(15,24)(17,20)(18,21)(19,22)(25,27)(26,31)(28,35)(29,34)(30,33)(32,39)(36,37)(38,40);(9,10)(11,20)(12,16)(13,17)(14,18)(15,19)(21,23)(22,24)(25,26)(27,31)(28,32)(29,30)(33,34)(35,39)(36,38)(37,40);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,12)(14,22)(15,21)(16,20)(18,24)(19,23)(25,28,29,36)(26,32,30,38)(27,35,34,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^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, 7714037);