#8900117

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

4438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=g,e^2=gi,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, 8807313);