#7973918

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	#44450	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4430 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)(9,10,15,17)(11,19,20,16)(12,22,21,18)(13,23,24,14)(25,26,29,32)(27,30,34,38)(28,31,36,40)(33,37,35,39);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,14)(13,18)(15,20)(17,19)(21,23)(22,24)(25,27)(26,30)(28,35)(29,34)(31,39)(32,38)(33,36)(37,40);(1,3,5,4)(2,7,6,8)(10,17)(11,20)(12,21)(13,24)(25,27,29,34)(26,30,32,38)(28,33,36,35)(31,37,40,39);(1,3,5,4)(2,7,6,8)(9,12,15,21)(10,18,17,22)(11,14,20,23)(13,19,24,16)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,39)(36,40);(1,4,5,3)(2,8,6,7)(9,13,15,24)(10,14,17,23)(11,18,20,22)(12,19,21,16)(25,27)(26,30)(28,35)(29,34)(31,39)(32,38)(33,36)(37,40);(9,14)(10,13)(11,21)(12,20)(15,23)(16,22)(17,24)(18,19)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=g,e^2=g,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, 7881114);