#7941749

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	#41081	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4305 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7848945);