#8291681

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

4415 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=hi,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8198877);