#8077410

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	#47931	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4484 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)(10,15)(12,17)(16,23)(20,24)(25,26,32,35)(27,36,39,33)(28,38,29,34)(30,37,40,31);(1,2,4,5)(3,6,7,8)(9,10,14,15)(11,17,19,12)(13,16,22,23)(18,24,21,20)(25,27,32,39)(26,33,35,36)(28,31,29,37)(30,38,40,34);(10,15)(12,17)(16,23)(20,24)(25,28,32,29)(26,34,35,38)(27,37,39,31)(30,33,40,36);(1,3,4,7)(2,6,5,8)(9,11,14,19)(10,12,15,17)(13,21,22,18)(16,24,23,20)(25,29,32,28)(26,34,35,38)(27,31,39,37)(30,33,40,36);(9,12)(10,11)(13,20)(14,17)(15,19)(16,18)(21,23)(22,24)(25,30)(26,31)(27,38)(28,36)(29,33)(32,40)(34,39)(35,37);(9,13)(10,16)(11,18)(12,20)(14,22)(15,23)(17,24)(19,21)(25,31)(26,30)(27,29)(28,39)(32,37)(33,38)(34,36)(35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=g,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, 7984606);