#7912286

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	#52129	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4309 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)(10,14)(11,17)(16,22)(18,19)(25,26)(27,33)(28,34)(29,39)(30,36)(31,37)(32,38)(35,40);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,27)(26,33)(28,38)(29,36)(30,39)(31,40)(32,34)(35,37);(9,10,13,14)(11,15,17,21)(12,16,20,22)(18,23,19,24)(25,28,32,33)(26,34,38,27)(29,30,40,37)(31,39,36,35);(9,10)(11,15)(12,16)(13,14)(17,21)(18,23)(19,24)(20,22)(25,29,32,40)(26,35,38,39)(27,36,34,31)(28,30,33,37);(1,3,4,7)(2,6,5,8)(9,11,13,17)(10,15,14,21)(12,19,20,18)(16,24,22,23)(25,30)(26,36)(27,39)(28,40)(29,33)(31,38)(32,37)(34,35);(9,12,13,20)(10,16,14,22)(11,18,17,19)(15,23,21,24)(25,31,32,36)(26,37,38,30)(27,40,34,29)(28,39,33,35)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=hi,e^2=g,f^2=i,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, 7819482);