#7876820

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	#43110	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4458 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=g,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7784016);