#7888966

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
prime radical	#59824	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4323 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)(11,13)(16,18)(19,21)(23,24)(25,26,31,28)(27,32,30,33)(29,36,38,34)(35,40,37,39);(1,3,5,8)(2,6,4,7)(9,10,15,12)(11,18,13,16)(14,17,22,20)(19,24,21,23)(25,27,31,30)(26,32,28,33)(29,35,38,37)(34,39,36,40);(1,2,5,4)(3,6,8,7)(9,11,15,13)(10,16,12,18)(14,19,22,21)(17,23,20,24)(25,27,31,30)(26,33,28,32)(29,35,38,37)(34,40,36,39);(9,12,15,10)(11,16,13,18)(14,20,22,17)(19,23,21,24)(25,28,31,26)(27,33,30,32)(29,36,38,34)(35,40,37,39);(1,2,5,4)(3,6,8,7)(9,13,15,11)(10,16,12,18)(14,19,22,21)(17,24,20,23)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40);(1,4,5,2)(3,7,8,6)(9,14)(10,17)(11,19)(12,20)(13,21)(15,22)(16,23)(18,24)(25,30,31,27)(26,33,28,32)(29,37,38,35)(34,40,36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7796162);