#8425039

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

4544 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332235);