#7844558

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

4366 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7751754);