#8091746

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
prime radical	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4332 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)(11,16)(12,17)(13,18)(14,15)(19,24)(20,21)(22,23)(26,28)(31,33)(32,36)(39,40);(1,2,4,5)(3,6,7,8)(25,26,30,28)(27,33,35,31)(29,32,38,36)(34,40,37,39);(10,15)(13,16)(17,23)(21,24)(26,28)(31,33)(32,36)(39,40);(1,3,4,7)(2,6,5,8)(9,11,14,18)(10,16,15,13)(12,20,22,19)(17,21,23,24)(25,27,30,35)(26,31,28,33)(29,37,38,34)(32,40,36,39);(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,24)(18,20)(25,28,30,26)(27,31,35,33)(29,36,38,32)(34,39,37,40);(9,13,14,16)(10,18,15,11)(12,21,22,24)(17,20,23,19)(25,29,30,38)(26,32,28,36)(27,34,35,37)(31,39,33,40)

Polycyclic presentation:

2^9|b^2=g,d^2=gh,e^2=i,f^2=hi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,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, 7998942);