#8891582

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

4414 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)(11,19,21,16)(12,17,22,20)(13,23,24,18)(25,26,29,32)(27,30,35,33)(28,31,36,38)(34,37,40,39);(1,2)(3,6)(4,5)(7,8)(10,15)(12,22)(16,19)(18,23)(25,26)(27,30)(28,31)(29,32)(33,35)(34,37)(36,38)(39,40);(2,5)(6,8)(10,15)(13,24)(16,19)(17,20)(28,36)(31,38)(34,40)(37,39);(3,7)(6,8)(9,11,14,21)(10,16,15,19)(12,23,22,18)(13,20,24,17)(25,26)(27,33)(28,31)(29,32)(30,35)(34,39)(36,38)(37,40);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,18)(13,19)(14,22)(15,20)(16,24)(21,23)(25,27)(26,30)(28,34)(29,35)(31,37)(32,33)(36,40)(38,39);(1,2)(3,6)(4,5)(7,8)(9,13,14,24)(10,18,15,23)(11,20,21,17)(12,19,22,16)(25,28,29,36)(26,31,32,38)(27,34,35,40)(30,37,33,39)

Polycyclic presentation:

2^9|a^2=g,d^2=h,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, 8798778);