#8179709

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

4508 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=h,e^2=h,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, 8086905);