#9607196

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

4237 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9514392);