#8471789

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

4501 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8378985);