#7941695

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

4434 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7848891);