#8418168

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

4749 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)(3,7)(4,6)(5,8)(9,10,14,15)(11,16,21,23)(12,13,18,17)(19,24,22,20)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(1,2,4,6)(3,5,8,7)(10,15)(12,18)(16,23)(19,22)(25,26,29,32)(27,33,35,30)(28,31,36,38)(34,39,40,37);(1,2)(3,5)(4,6)(7,8)(10,15)(13,17)(16,23)(20,24)(28,36)(31,38)(34,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,11,14,21)(10,16,15,23)(12,22,18,19)(13,20,17,24)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(9,12,14,18)(10,17,15,13)(11,19,21,22)(16,20,23,24)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(9,13,14,17)(10,18,15,12)(11,20,21,24)(16,19,23,22)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=h,e^2=h,f^2=gi,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, 8325364);