#7719178

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	#48820	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4235 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7626374);