#8181158

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

4582 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8088354);