#7899246

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

4509 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)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26)(27,30)(28,33)(29,32)(31,36)(34,38)(35,40)(37,39);(1,2,4,5)(3,6,7,8)(25,26,30,27)(28,31,36,33)(29,32,38,34)(35,39,37,40);(10,11)(15,17)(16,18)(22,23)(26,27)(31,33)(32,34)(39,40);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,27,30,26)(28,33,36,31)(29,34,38,32)(35,40,37,39);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,28,30,36)(26,31,27,33)(29,37,38,35)(32,40,34,39);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29,30,38)(26,32,27,34)(28,35,36,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=h,e^2=g,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, 7806442);