#8069343

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

4396 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=gh,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7976539);