#8097726

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

4523 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004922);