#8891781

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

4355 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8798977);