#8041187

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

4376 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,e^2=gh,f^2=g,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, 7948383);