#7888937

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

4491 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=h,e^2=gi,f^2=hi,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, 7796133);