#7987880

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	#36442	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4452 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)(11,17)(12,14)(13,16)(15,18)(19,23)(20,21)(22,24)(25,26,31,28)(27,32,30,34)(29,36,38,33)(35,40,37,39);(1,3,5,8)(2,6,4,7)(9,10,14,12)(11,15,18,17)(13,20,21,16)(19,23,24,22)(25,27,31,30)(26,32,28,34)(29,35,38,37)(33,39,36,40);(1,2,5,4)(3,6,8,7)(10,12)(11,18)(16,20)(19,24)(26,28)(32,34)(33,36)(39,40);(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,28,31,26)(27,34,30,32)(29,36,38,33)(35,40,37,39);(1,2,5,4)(3,6,8,7)(9,12,14,10)(11,17,18,15)(13,16,21,20)(19,22,24,23)(25,29)(26,33)(27,35)(28,36)(30,37)(31,38)(32,39)(34,40);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,30,31,27)(26,34,28,32)(29,37,38,35)(33,40,36,39)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=hi,e^2=g,f^2=ghi,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, 7895076);