#7898147

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4047 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=gh,e^2=i,f^2=i,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, 7805343);