#8417289

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

4372 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=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, 8324485);