#7893797

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

4247 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=hi,e^2=i,f^2=hi,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, 7800993);