#8139660

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

4600 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8046856);