#8509502

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

4488 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)(11,19)(12,22)(13,18)(14,16)(15,21)(17,23)(20,24)(25,26,29,32)(27,30,34,38)(28,31,36,35)(33,37,40,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,21)(10,15,16,19)(12,20,23,18)(13,17,24,22)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40);(2,5)(6,8)(10,16)(12,23)(15,19)(18,20)(25,27,29,34)(26,30,32,38)(28,33,36,40)(31,37,35,39);(3,7)(6,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,19)(17,24)(18,23)(25,26)(27,30)(28,35)(29,32)(31,36)(33,39)(34,38)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,20)(13,19)(14,23)(15,24)(16,22)(18,21)(25,27)(26,30)(28,33)(29,34)(31,37)(32,38)(35,39)(36,40);(1,3)(2,6)(4,7)(5,8)(9,13,14,24)(10,18,16,20)(11,17,21,22)(12,19,23,15)(25,28,29,36)(26,31,32,35)(27,33,34,40)(30,37,38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416698);