#7852768

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

4503 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7759964);