#7931623

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

4295 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=ghi,d^2=ghi,f^2=hi,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, 7838819);