#7714571

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

4376 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)(11,16)(12,14)(13,22)(15,17)(18,24)(19,23)(20,21)(29,37)(33,38)(34,40)(36,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,13)(10,16,17,22)(12,21,18,19)(14,20,24,23)(25,26)(27,28)(29,33)(30,31)(32,35)(34,36)(37,38)(39,40);(1,3,4,7)(2,6,5,8)(10,17)(11,13)(12,18)(20,23)(26,31)(28,32)(33,38)(36,39);(9,12,15,18)(10,14,17,24)(11,19,13,21)(16,23,22,20)(25,27)(26,32)(28,31)(29,34)(30,35)(33,39)(36,38)(37,40);(9,13,15,11)(10,16,17,22)(12,21,18,19)(14,23,24,20)(25,28,30,32)(26,27,31,35)(29,36,37,39)(33,34,38,40);(9,14)(10,18)(11,20)(12,17)(13,23)(15,24)(16,19)(21,22)(25,29)(26,33)(27,34)(28,36)(30,37)(31,38)(32,39)(35,40)

Polycyclic presentation:

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

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, 7621767);