#8041157

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

4541 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=g,e^2=gh,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, 7948353);