#7973038

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	#48149	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4559 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7880234);