#7912230

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	#49859	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4251 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)(11,15)(12,20)(13,18)(14,16)(17,22)(19,24)(21,23)(25,26)(27,30)(28,34)(29,37)(31,33)(32,36)(35,38)(39,40);(1,2,4,5)(3,6,7,8)(9,11,14,13)(10,15,16,18)(12,19,22,21)(17,23,20,24)(25,27,32,31)(26,33,36,30)(28,37,39,35)(29,34,38,40);(10,16)(15,18)(17,20)(23,24)(25,28,32,39)(26,34,36,40)(27,35,31,37)(29,30,38,33);(9,10)(11,15)(12,17)(13,18)(14,16)(19,23)(20,22)(21,24)(25,29,32,38)(26,35,36,37)(27,34,31,40)(28,30,39,33);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,19)(13,21)(14,22)(15,23)(16,20)(18,24)(25,30)(26,27)(28,38)(29,39)(31,36)(32,33)(34,35)(37,40);(9,13,14,11)(10,18,16,15)(12,21,22,19)(17,24,20,23)(25,31,32,27)(26,33,36,30)(28,37,39,35)(29,40,38,34)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=i,e^2=gi,f^2=hi,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, 7819426);