#8057851

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

4280 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

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, 7965047);