#8097257

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

4374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=g,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, 8004453);