#7906004

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

4483 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)(9,10,15,17)(11,16,20,22)(12,13,21,18)(14,19,23,24)(25,26,31,28)(27,34,30,32)(29,36,38,33)(35,39,37,40);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,14)(13,19)(15,20)(17,22)(18,24)(21,23);(10,17)(11,20)(12,21)(19,24)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,3,5,4)(2,7,6,8)(9,12)(10,18)(11,14)(13,17)(15,21)(16,24)(19,22)(20,23)(25,28,31,26)(27,32,30,34)(29,36,38,33)(35,39,37,40);(1,3,5,4)(2,7,6,8)(9,13,15,18)(10,12,17,21)(11,19,20,24)(14,22,23,16)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,4,5,3)(2,8,6,7)(9,14)(10,19)(11,12)(13,16)(15,23)(17,24)(18,22)(20,21)(25,30,31,27)(26,32,28,34)(29,37,38,35)(33,39,36,40)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=ghi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7813200);