#7974951

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

4503 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,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, 7882147);