#7883085

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4311 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)(9,10)(11,12)(13,18)(14,16)(15,17)(19,20)(21,24)(22,23)(25,26,31,34)(27,35,30,32)(28,29,33,39)(36,40,38,37);(1,2,4,5)(3,6,7,8)(9,11,14,17)(10,15,16,12)(13,19,22,24)(18,21,23,20)(25,27,31,30)(26,32,34,35)(28,36,33,38)(29,37,39,40);(10,16)(11,17)(18,23)(19,24)(25,26,31,34)(27,35,30,32)(28,29,33,39)(36,40,38,37);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,17)(13,18)(14,16)(19,21)(20,24)(22,23)(25,28,31,33)(26,29,34,39)(27,36,30,38)(32,37,35,40);(9,12,14,15)(10,17,16,11)(13,21,22,20)(18,19,23,24)(25,29)(26,33)(27,37)(28,34)(30,40)(31,39)(32,38)(35,36);(9,13,14,22)(10,18,16,23)(11,19,17,24)(12,20,15,21)(25,30,31,27)(26,32,34,35)(28,38,33,36)(29,37,39,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=gh,e^2=h,f^2=i,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, 7790281);