#8096555

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=ghi,f^2=ghi,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, 8003751);