#7732474

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

4423 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=gh,e^2=i,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7639670);