#8416807

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

4561 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)(11,17)(15,18)(19,21)(23,24)(25,26)(27,32)(28,31)(29,34)(30,33)(35,39)(36,38)(37,40);(1,3,5,8)(2,6,7,4)(9,10)(11,18)(12,20)(13,14)(15,17)(16,22)(19,23)(21,24)(25,27)(26,32)(28,34)(29,31)(30,35)(33,39)(36,40)(37,38);(1,3)(2,4)(5,8)(6,7)(10,14)(15,18)(16,20)(23,24)(25,26,31,28)(27,34,29,32)(30,36,38,33)(35,39,37,40);(1,4)(2,3)(5,6)(7,8)(9,11,13,17)(10,15,14,18)(12,21,22,19)(16,24,20,23)(25,28,31,26)(27,34,29,32)(30,36,38,33)(35,40,37,39);(9,12,13,22)(10,16,14,20)(11,19,17,21)(15,23,18,24)(25,29,31,27)(26,32,28,34)(30,37,38,35)(33,39,36,40);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,30,31,38)(26,33,28,36)(27,35,29,37)(32,39,34,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=gi,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, 8324003);