#7883097

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

4267 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)(11,16)(12,18)(13,23)(14,20)(15,17)(19,22)(21,24)(25,26,31,29)(27,34,30,32)(28,33,38,36)(35,40,37,39);(1,2,5,6)(3,7,4,8)(9,11)(10,16)(12,22)(13,20)(14,23)(15,21)(17,24)(18,19)(25,27,31,30)(26,32,29,34)(28,35,38,37)(33,39,36,40);(10,17)(11,21)(12,20)(19,23)(25,26,31,29)(27,34,30,32)(28,33,38,36)(35,40,37,39);(1,3,5,4)(2,7,6,8)(9,12,15,20)(10,18,17,14)(11,22,21,13)(16,19,24,23)(25,28,31,38)(26,33,29,36)(27,35,30,37)(32,39,34,40);(1,4,5,3)(2,8,6,7)(9,13,15,22)(10,19,17,23)(11,20,21,12)(14,16,18,24)(25,29,31,26)(27,34,30,32)(28,36,38,33)(35,40,37,39);(9,14,15,18)(10,20,17,12)(11,23,21,19)(13,24,22,16)(25,30,31,27)(26,32,29,34)(28,37,38,35)(33,39,36,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=gh,e^2=gh,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, 7790293);