#7903853

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

4456 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:

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

Polycyclic presentation:

2^9|c^2=ghi,d^2=i,f^2=h,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, 7811049);