#7734356

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	C₂×#535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4588 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)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(25,26,30,32)(27,28,33,31)(29,34,38,40)(35,36,37,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,17)(20,24)(22,23)(25,27,30,33)(26,31,32,28)(29,37,38,35)(34,36,40,39);(11,18)(12,20)(14,16)(15,22)(26,32)(27,33)(34,40)(35,37);(1,3,4,7)(2,5,6,8)(9,12,13,20)(10,15,17,22)(11,19,18,24)(14,21,16,23)(25,28,30,31)(26,33,32,27)(29,36,38,39)(34,37,40,35);(9,11)(10,16)(12,19)(13,18)(14,17)(15,23)(20,24)(21,22)(25,26)(27,31)(28,33)(29,34)(30,32)(35,39)(36,37)(38,40);(9,10)(11,16)(12,15)(13,17)(14,18)(19,23)(20,22)(21,24)(25,29)(26,34)(27,35)(28,36)(30,38)(31,39)(32,40)(33,37)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7641552);