#8209864

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

4584 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:

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

Polycyclic presentation:

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

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, 8117060);