#8271919

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

4638 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)(3,4)(5,6)(7,8)(10,11)(15,17)(16,18)(22,23)(25,26,30,32)(27,35,37,31)(28,38,39,33)(29,34,40,36);(2,6)(3,7)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,27,30,37)(26,31,32,35)(28,34,39,36)(29,33,40,38);(1,3,5,7)(2,4,6,8)(13,21)(16,18)(19,24)(22,23)(26,32)(29,40)(31,35)(33,38);(1,4)(2,7)(3,6)(5,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,28,30,39)(26,33,32,38)(27,36,37,34)(29,35,40,31);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,27)(26,31)(28,36)(29,33)(30,37)(32,35)(34,39)(38,40);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29)(26,34)(27,33)(28,35)(30,40)(31,39)(32,36)(37,38)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=gi,e^2=ghi,f^2=ghi,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, 8179115);