#7896079

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4398 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)(9,10,14,17)(11,12,19,15)(13,16,21,23)(18,20,24,22)(28,36)(31,38)(34,40)(37,39);(1,3,5,8)(2,6,4,7)(25,26)(27,32)(28,31)(29,33)(30,35)(34,39)(36,38)(37,40);(9,11,14,19)(10,15,17,12)(13,18,21,24)(16,22,23,20)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,24)(22,23)(25,28,29,36)(26,31,33,38)(27,34,35,40)(30,37,32,39);(9,12)(10,11)(13,20)(14,15)(16,18)(17,19)(21,22)(23,24)(25,27)(26,32)(28,34)(29,35)(30,33)(31,39)(36,40)(37,38);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,16,17,23)(11,18,19,24)(12,20,15,22)(25,26)(27,30)(28,31)(29,33)(32,35)(34,37)(36,38)(39,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=i,f^2=gh,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, 7803275);