#8044547

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	#42484	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4586 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7951743);