#8057900

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

4319 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=gi,f^2=hi,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, 7965096);