#8044527

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	#47840	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4389 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)(25,26,31,33)(27,30,28,32)(29,34,38,40)(35,39,36,37);(1,3,5,8)(2,6,4,7)(11,17)(15,21)(18,23)(22,24)(25,27,31,28)(26,32,33,30)(29,36,38,35)(34,39,40,37);(1,2,5,4)(3,6,8,7)(10,14)(15,21)(16,19)(22,24)(26,33)(27,28)(34,40)(35,36);(1,2,5,4)(3,6,8,7)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28,31,27)(26,30,33,32)(29,36,38,35)(34,39,40,37);(1,4,5,2)(3,7,8,6)(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,29)(26,34)(27,35)(28,36)(30,39)(31,38)(32,37)(33,40);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,30)(26,27)(28,33)(29,37)(31,32)(34,36)(35,40)(38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=g,e^2=gh,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, 7951723);