#7735613

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

4445 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)(25,26)(27,33)(28,31)(29,34)(30,32)(35,40)(36,38)(37,39);(1,2)(3,5)(4,6)(7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,27,30,28)(26,31,32,33)(29,35,37,36)(34,38,39,40);(10,14)(11,17)(16,19)(18,23)(26,32)(31,33)(34,39)(38,40);(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,40)(37,39);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,28,30,27)(26,33,32,31)(29,36,37,35)(34,40,39,38);(1,3,4,7)(2,5,6,8)(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)(25,29,30,37)(26,34,32,39)(27,35,28,36)(31,38,33,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=i,e^2=h,f^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, 7642809);