#7714791

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	#40442	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4143 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7621987);